The problem in question:
Telekinetic movement states:
You can move an object or a creature with your mind. As an action, you target one loose object that is Large or smaller or one willing creature, other than yourself. If you can see the target and it is within 30 feet of you, you can move it up to 30 feet to an unoccupied space you can see. Alternatively, if it is a Tiny object, you can move it to or from your hand. Either way, you can move the target horizontally, vertically, or both.
Let's look at certain cases here:
Say, a Huge size carriage is drawn onto a wooden planks nailed together. This plank construction is size large. Can you use this feature to move the construction up, also moving up the carriage?
A hook is on top of the carriage, nailed to it. Can you use this feature to move tiny metal ring to hook up on the hook and elevate the carriage?
How, if at all, would the answer change if a creature of huge size woud be a subject of those shenanigans?