I have a small concrete patio outside my back door. Over the last few years it appears to be moving slowly away from the house creating a 1"-2" gap. The patio is ringed by several feet of paver stone so I don't expect it to shift very far. I wanted to fill the gap with crushed limestone (for safety and aesthetic reasons) but my wife feels it will push the patio out or up. I have the same issue with some of the expansion joints in my driveway and I would like to do the same thing.

Limestone screening should be fine, but it will also just keep moving.

I would use some Butyl caulk to keep the water out of that crack so water can't freeze in that gap area and keep pushing the patio out.

Same thing for driveway.. Butyl caulk

You can use stone etc to pre fill some of the gap up and not use so much caulk.