My question basically concerns summoned objects such as a Spiritual Weapon or an Echo Knight's Echo, which can "float" in the air, and terrain moving under the space it occupies. Do these objects move with the moving terrain? 

Two examples based on some recent games I am a part of

1) I am playing a Cleric in a pirate-themed campaign. I summoned my spiritual weapon on a nearby ship to attack the enemy crew. This ship continued moving on its own turn. Does my spiritual weapon move with the ship (occupying the same relative space to the creature I was attacking) or does it continue floating in the same space and effectively move away from the ship (and move away from the enemy)?  

2) My brother is playing an Echo Knight in my Out of the Abyss campaign. He summoned his echo in the basket of a mechanical lift, which he then lowered. Does the echo move downward with the basket or does it stay floating in the space he moved it to as the basket lowers? 

Both of these events were resolved in a satisfying way for those games, but it does leave me wondering about the exact mechanics of this sort of thing. While its easy enough to rule these one way or another, I am wondering if there is any definitive RAW text that states whether either or both of these would interact with the environment one way or another

The way I read it, by RAW, they don’t move with the ship. For example with spiritual weapon, ”you can move the weapon” to me says that it’s otherwise stationary. It’s not like an enemy can grapple it and drag it around. The thing only moves when you tell it to, and that can only happen on your turn, when you spend a bonus action making it do so.

However, in play, I would probably let them maintain position relative to the ship, because it’s just much simpler to track where it is, and not worry about how fast the ship is moving or if it’s turned a bit, or if a wave has come through and the deck is now 12’ higher than it was last round.

You cast the Spiritual Weapon spell and it promptly wooshes away at 1 million miles per hour because of the spin of the galaxy.

The way I think of it is that a ship such as those found in Ghosts of Saltmarsh has its own map, with its own squares, so anything in those squares moves with the ship unless the DM decides otherwise. In other words you treat the ship like a static map, almost as if everything else is moving around the ship instead. This is especially appropriate for Spelljammer where the ships actually do have their own gravity and air bubble.

Personally the only exceptions I'd be likely to make would be for flying creatures, to encourage them to land or perch somewhere if they want to stay with a ship, otherwise they're assumed to be strafing across it and need to use some of their movement to keep up. Depends heavily on how complicated that might make things though, and the cinematic aspect as well.

For example if I were running a battle between a moving ship and a dragon, and you want it to circle in a cool way, actually accounting for the distance would mean it might need to keep taking the Dash action and losing its turn, while the ship defenders pelt it with long ranged attacks (long ranged spells, sharpshooter longbows, ballistae etc.) which would make the fight stupidly easy. So I might just narrate it instead, unless I'm purposefully giving the party a chance against a dragon that's otherwise too high level for them, e.g- just have it circle then strafe with its breath weapon, but giving the players several rounds to shoot back, heal up and prepare for the next pass.

Very much a case-by-case basis IMO. For a wagon chase I would probably count them as being one or two squares and functioning as vehicles in the same way, but might account for speed when deciding how easy a jump between two wagons might be etc.

The weapon can only move 20 feet at a time so yes the boat would move out from under it unless you used the weapons movement to keep it in a relative place to the boat.

If you cast it on a flying carpet could you add the carpets movement to the spiritual weapons movement? If not why would you add the movement of the boat?

I've seen it ruled both ways. I think the most satisfying answer is that they move relative to their environment (what is "stationary" really?).

But it is up to DM. The RAW doesn't have a clear answer.

Thematically and for the sake of gameplay, it is easier to have the spiritual items maintain position on the "boat map" or whatever it is that is moving. Just like it is easier to say the deck of the ship is the "ground" for targeting or description purposes of spells and items.