All, I had an idea pop in my head this morning, which was "How would I keep characters from getting out of chains/cells?  My party is 11th level (may be higher when this occurs, if it ever does) and consists of a rogue assassin, a blood hunter (order of the lycan - tiger) and a transmuter mage. First all, how do you keep the mage from using spells without using anti-magic. I have already done that a few times and don't want to repeat it. I was thinking a mask with a gag in and bound hands, plus no access to normal spell components. Secondly, if the blood hunter is changed and goes hybrid, would it be reasonable to assume the cuffs would slip off? And third, for the rogue, what would you allow them to scrounge up to make makeshift thieves tools? Assume they are in a mine, in the hold of a ship, or in a cell.

Depends on how long you need the party's capture to last; it's basically impossible to keep a high-level caster imprisoned for longer than maybe a few days without using anti-magic. Eventually the caster is going to need water, and when the gag comes off, there's a wide range of teleportation spells that only require Verbal components (some of them, like Dimension Door, even let you bring friends along).

I'm not especially familiar with Blood Hunters, but reviewing their abilities real quick, I don't see any reason the Lycan transformation should automatically remove their restraints. You can use armor and equipment while transformed and there's no mention of their size changing, so I think it's reasonable to say the cuffs stay on. Now, whether the Hybrid form should be able to break the cuffs is up to you.

I don't know if I would let a Rogue improvise Thieves' Tools out of what they can find in a prison cell, but I would say slipping out of cuffs is probably a Sleight of Hand check, so if they've got high Dex and nothing else to do, they'll get it eventually.

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All, I had an idea pop in my head this morning, which was "How would I keep characters from getting out of chains/cells?"

Important questions for scenarios like this are "do you want the PCs to escape?" and "why does the enemy want them alive?". In general the reliable way to keep PCs contained is to have them entirely incapacitated -- for example, cast flesh to stone on them before transport -- but that's not much fun.

GM: The guard captain stands in front of you. Behind her is a large man wearing the traditional garb of the executioner. He holds a hatchet and a pair of pliers. Beside the executioner stands a small man in the robes of the state religion.

GM: The captain speaks. "The priest behind me is a cleric of Pelor. He knows spells to regenerate missing tongues and fingers. Are you going to behave or are we going to need the cleric's spells?"

In a world with magic, it would be understood that in order to jail a caster, you need special manacles that bind the fingers as well as the hands (and feet/toes as well) to hinder somatic casting, as well as gags to keep from verbally casting a spell. 

For class abilities that aren't technically spells, it depends on how unique your characters' abilities are. Are there NPC transmuter mages? Is that a common thing to be? If so, someone at some point will have needed to imprison one and may have used a super dense metal that's resistant to transmutation magic or something to fashion the chains out of.  Otherwise, if the player is more special and uniquely gifted with the powers of  transmutation wizard and player classes are beyond the scope of normalcy in your world, then it's likely there would be no such counter unless they were taken by someone specifically after the player and armed with knowledge of their powers and their limitations. 

For hindering a caster without using anti-magic fields:

A Silence spell would prevent the use of verbal components.  There's also the Blindness/Deafness spell or just a simple blindfold (or a bag over the head) to prevent the use of any spell that requires the caster to see their target or the spell's target location-- although the spell gives a saving throw every turn, so it's not a long-term possibility.  If this is taking place in a prison where the guards could reasonably be prepared to hold a caster, they could have someone standing by to cast Counterspell if needed.  A Geas spell could be used to inflict psychic damage if the targeted character attempts to cast any spells (although someone would have to successfully cast it on the PC first).  A Glyph of Warding in the caster's cell could be set to trigger if the cell's occupant tries to cast a spell.  

While this is entirely homebrew, I have an 'anti-magic' metal that drains magic away like a lightning rod. It's rare and expensive, so it's not just something you can pop down to the market and buy, and it's essentially useless in combat (it requires engineering to work), but it will neatly drain away magical ability and deny regaining such. 

And then you just seal the door with Stone Shape. Or similar. I mean, if you really want to crush their hopes of escape. 

The facetious answer to "how do you keep a wizard from casting" is strap a shield to their arm. A non-facetious answer can be extrapolated from the observation that equipping non-proficient armor prevents spellcasting. Provided the baddies have enough information to know about the spellcaster's capabilities, they could easily have an artificer create something that counts as heavy armor and includes a locking mechanism... The bigger problem is how to prevent the transmuter from changing the, presumably, iron restraints into wood or copper/silver.

So far as the cosmetics of the bloodhunter's hybrid form... It's entirely subjective. Maybe they get bigger and the restraints burst. Maybe they get smaller and they slip off. Or maybe they don't get enough bigger or smaller and the restraints still fit. The issue is, what have you and/or the player described in the past, because you ought to be consistent... Makeshift thieves tools often can be a dagger. But mine/ship/cell makes me think of an iron nail--one of the big old wrought iron ones.

I recently ran a "you've been captured" scenario. For my money the thing you really need to think about is gear. Where did the gear the characters had go, and if that's no longer accessible, how will they be able to collect needed items like rope and tools. Also, weapons and armor. One of the things I had to accomplish was to get a weapon of warning gone for good--so some of their things had to be inaccessible, but that meant I needed to be able to offer new items in the place of old ones. Good luck!

In high magic worlds - I don't explain. I simply say that this prison and/or cell was constructed in such a way that casting and utilising of abilities is entirely fruitless. People far more skilled than adventurers made this place and it is simply inescapable. Now let's montage the rest of the time in prison until the sentance is served. You've all lost levels appropriate to the amount of time you spent inside.

In low-magic worlds - again, I don't bother. Having played across dozens of worlds and game systems there is only one conclusion for criminal adventurers in a low-magic setting. For them, the punishment is death or exile which effective ends the journey of those characters. 

The answer simply put is to not explain it beyond - this is what happens and your abilities don't work here.

Now, if you're running an adventure set or starting in a prison, yeah that's a little more difficult. It has to be at low level and without the feats optional rule, also limiting sources to Xanathar's, Basic Rules and Player's Handbook spells. With this, the only spells you have to worry about at low level (Cantrips and Level 1) are Vicious Mockery and Dissonant Whispers which can be taken care of with a simple high wisdom on the part of the guards. As long as the prisoners are manacled, they are restrained and unable to attack. Treat a command to approach or give keys or unlock as something harmful to the guards (because it would be) and Bob's your uncle. Of course, I am treating having hands manacled to a belt or some such as Incapacitated because the writers really messed up the Restrained condition. I mean if you've got hands bound behind your back stands to reason you're not casting using somatics doesn't it?

Even if you branch out to Tasha's - which is to my mind the source book that began to make the DMs job that much more complex and scenarios more limited - the only spells we're really concerned about doing damage to the guards are Lightning Lure, and Sword Burst. In which case the answer is to have wide hallways outside the cells and food is only delivered to prisoner via a chute, or if the prisoner steps away from the door far enough. The locks are needless to say magical.

Once characters achieve a second level spell slot and have access to Thunder Step or Misty Step, yeah that's the point at which a prison needs to have countermeasures. The simple answer is a door permanantly enchanted to maintain a field of silence over the cell, and have the prisoner's hands restrained such that somatics aren't possible. They won't have material components so spellcasting at that point is pointless. 

As for getting through the door they need to beat an AC of 23 or even higher (for a reinforced Adamantine door). Even if they do, you have the door warded such that it's a deadly trap dealing appropriate damage for their level. So even if they can unarmed strike the door, they're gonna wish they hadn't.

TL;DR here though is that if you're planning a campaign starting while in prison then simple limit sources to just Player's Handbook and Basic Rules, keep the play at low level. If you're already playing and the player characters have earnt themselves jail time either say that in this prison using abilities isn't possible because of yada yada, or have them on the run forever and those character's adventures end should they escape. 

Do you want them to not escape, or do you want them to have to work for it? (And fail if they're not clever.)

Obviously, everyone's going to be familiar with casters, and they know to take away their tools, bind their hands, and gag them. That'll stop everyone but a sorcerer, and it ought to stop the transmuter's subclass ability.

But the more obscure their powers, the more their enemies have to know explicitly about them to stop them, and most non-spell abilities can simply be used at will.

Or ... permanent silence, darkness, hand restraints. No verbal components, no somatic components, no material components, no line of sight.