That's a good question. I don't know...but my gut says the same thing.

Carol

I always leave the newest bin on and add above it. In your case I would harvest the bottom two and leave the top as new bottom. The bin is already established so you won't need to add the sheet of newspaper as you would if you were starting over. I also like to pile up the harvested compost and skim the top every once in a while to drive any worms down that may be still in there. Then just add the very last bit when it's mostly worms back to the new bin to kick it off.

So without the newspaper, will the worms still stay in the bin and not fall into the bottom?

Material that's been in the bin for a while and has a lot of castings in it naturally clumps together in a mass. So if you've got a tray of material that's been going for a while, the worms and bedding won't fall out the mesh at the bottom.

If you pick up a tray and find worms hanging out of the mesh, they're not falling, they've just been interrupted mid-commute.