I'm about to finish Dragon Heist and I'm considering jumping into my own homebrew campaign setting.  My players have expressed interest in keeping their characters, but I don't really want to just say, 'Hey, we're moving to a different setting!'  Anyone have any fun ideas for why and/or how the PCs would get shifted to another setting?  

How about some kind of world-shifting planar-travel type thing?

Maybe some BBEG has a McGuffin and the PCs chase him through a doorway or swirling vortex and... end up in another world. They beat him eventually but have no way to get back.

Or maybe this can lead into another series of adventures, with the ultimate goal being (at campaign's end) the option to go back home or stay in New World Land.

Have a mysterious stranger ask them for help and when they agree the NPC transports them to your realm.

Your PCs are in your world all the time, except they were in some stasis chambers, dreaming about a virtual city called Waterdeep. All their adventures in Waterdeep were some sort of tests and they passed it. Now they wake up in the "real" world, and are offered a job by the ones who "tested" them.

There’s a problem in the homebrew world, and someone is going around recruiting groups of people who seem like they could help, then plane shifting them there. 

Ooh, I like the "your past adventures were all a dream" thing. I'd probably go with that.

A really old-school way of doing it would be a cursed scroll.

I like the idea of having an adventure that winds up having them chase someone to the other world.

The only problem with the all a dream thing would be their gear. If they have any really cool stuff that wouldn’t transfer over to the new world. I’d say you can, of course, give them new stuff, but it should be different, not you happen to find a flaming sword on the locker. Hey, Bob, didn’t you used to have a flaming sword, that would feel cheesy. 

Quote from Fatso101 >>

Anyone have any fun ideas for why and/or how the PCs would get shifted to another setting?  

Unless your homebrew is radically different from the current setting, why not simply make it a separate continent? I'm putting Khorvaire on Toril with Faerun for similar reasons.

I can't find the reference, but someone said that IRL samurai, pirates, African tribespeople, and the industrial revolution all existed at the beginning of the 1800s when Napoleon was conquering Europe, so we know this type of world could exist in fiction.

So Dragon Heist, huh?

I have run and finished Dragon Heist. This is what I would recommend. 

Get your party involved with the Xanathar Guild again if they haven't been recently. If you already have them involved in it, or Xanathar was the villain, this may be a natural transition, but either way it shouldn't be that hard. Once the Xanathar Guild is involved again, have them be kidnapped, captured, or forced to meet the Xanathar. Once they meet, have the Xanathar either be neutral or hostile, but not immediately physically aggressive. He may start out rambling about his pet immortal goldfish, how he can kill them all, or how he can't dream, something like that.

After he starts talking, have find someway for him to be offended by him. Just wing it, he is insane, he will make up a reason or find something horribly offensive in their behavior. If they are silent trying not to offend him, their quietness could be offensive. If they call him sir, maybe he doesn't like that pronoun. If they said hi to him, maybe he gets insulted they didn't say hi to Sylgar. 

Anyway, have him suddenly try to shoot an eye ray, just one, at a random party member, not the one that offended him, but at one of them. This is where all hell breaks loose. Your players will undoubtedly be scared, offended, aggressive, so on. Start combat. 

After the party is bested, all paralyzed, petrified, disintegrated, dropped to 0 hit points or something like that, Xanathar calmly disintegrates all of their bodies. 

Now you reveal that Xanathar's lack of dreams has started to manifest in his disintegration eye ray. Any creature disintegrated by him is transported into his dream realm (insert homebrew campaign world's name). This is how your party gets to the world. You literally have it be the dream realm of a beholder, who's dreams can shape and create reality.

Thanks everybody!  A lot of solid options here.  Got a lot to think about.  Honestly, I'm leaning toward a quick jump.  I didn't really care for running Dragon Heist and I don't think my players much liked it either.  They're fairly used to my own worlds with a lot of open world exploration.  I only used Dragon Heist because I wanted something to run with minimal writing on my part.  But, I found it way less fun, so I'm taking on the 'burden' anyway.  

I'm thinking I might combine a couple options.  A god or something from my homebrew world will bring them there, but it'll seem like waking from a dream to them.  They'll get to keep all their stuff, including the building, but they'll be a little confused about the new world, but the world will treat them as if they've always been there.  I might even tweak some of the more interesting characters from Dragon Heist to mess with them a little more.

@LeviRocks, I really do like the idea of a beholder's dreams.  I might use that in my homebrew adventure anyway.  Solid idea.  Love hearing interesting twists like that.  

There are lots of options ...

1) You could respin Waterdeep as any large city in your homebrew world. I realize that you have already played it but if the players are onside and the city would fit then just rename some of the NPCs/factions and allow the players to keep the inn in a big city in your homebrew world as a home base.

2) Planar travel. Mirror or portal, request from a high ranking Waterdeep wizard, perhaps the characters are framed for a crime then forced to travel to a different plane to accomplish a mission or just get rid of them. There are a million possible ways to do it. However, unless your homebrew plot lines tend to involve gods, I would tend to opt for a more mundane option. Perhaps a wizard from that world has traveled to Waterdeep in search of just the right group of adventurers. Perhaps whatever task is needed to start the plot line can not be performed by a creature native to that plane of existence.

3) The dream option. I tend to prefer the options that logically transport the characters to the new realm with sufficient motivation since to me it offers a stronger feeling of continuity. However, respinning the Waterdeep adventure as a dream of some sort ... perhaps the characters were adventuring in your world, found a magical pool, not knowing what it did they experimented and found themselves living lives elsewhere but not recalling their own lives. Have something happen either at the end of the Waterdeep adventure or in their native world, perhaps an earthquake?, that interrupts the magic of the pool, they find themselves emaciated and exhausted but (if it fits your storyline) they could magically have all their gear. The pool was cursed, luring adventurers into experiencing an alternate reality, the party ends up barely escaping with their lives since the pool would have soon finished consuming their life force. Start your adventure from the cave, you have the option of the characters remembering who they were or having to discover their memories. Lots of cool options.

Did you ever explicitly state when you were running Dragon Heist that the city of Waterdeep was in a place called Ferun, on a world called Toril?

Because what you could do is just say there's a Waterdeep in your world, and they've been in your world all along. If you had referenced Ferun, you could always have Ferun be a small country with the city of Waterdeep in it.

Always feel free to just transplant locations into your homebrew world if you like them enough. "Good artists borrow, great artists steal." Pablo Picasso CharlesthePlant

Just move them. Not really a big deal.  Put them on a ship and take them across the ocean.