While 'they' are carrying in the upstairs furniture...

Is this a professional moving company then? They should have with them anything they need to protect, can you find out if that's true with your movers. Here, it was a sideways blowing rainstorm. Small local company hired to only move the larger pieces and they treated as if their own things.

They had something like painters drop cloth that was tacky on one side so it wouldn't slip around, cut that and laid it down hallways going to bedrooms etc (blond carpet that end of the house). Their footing was safe and so was my carpet - I would guess it would work on stairs but there are no stairs here.

What ever caused the 'tack' didn't seem to leave any residue but they had a big professional roll so I didn't see any product name.

Self adhering temporary carpet protection plastic is available at places like Lowe's. You've probably seen this stuff at model homes, etc.

I would just leave the paper and cardboard down as protection. If the movers don't bring something, throw down some old blankets or sheets to collect the water. Tape them down. Don't put the plastic near a doorway. It will be too slippery when wet.

I don't recall what my last movers used but we kept the paper/cardboard down. I think they also brought carpet/floor protectors since it was snowy. You can always tape the tarp down.