The moving crate vanishes leaving behind the gold boxes and anything else that was inside of it. All of this stuff (including any vendor bags with checks inside) are all piled on the same tile in the center of where the house was. They are piled on top of each other, so grab a bag or box and move it and there will be more until they are all gone. It's often a nice source of items, though most usually you will find stuff in there like plants (from customizing with plants on the front steps, they'll end up in the crate), or vines or messageboards.
It used to be you could demo a house with stuff in the crate, so people could fabricate gold boxes that way. Fill your crate by customizing over lots of boxes, drop house, get stuff, rinse and repeat. Probably why they changed it, hehe.