Yes, if it's compost, you can move it around, dig it in where it is, or whatever suits your fancy.

If it's not totally composted - for example, if you piled up a lot of leaves over your potatoes and they are still recognizable as leaves - it may not be a good idea to dig or till it in. But you could certainly move it and use it as mulch somewhere else, or add it to a compost pile to finish.

I agree with tox.

After compost has been applied in a few weeks or months I notice there are some left over piece of wood, leaves or sticks, so I gather all those up and put them back in the bin. Then I take some new compost out of bin and repeat the process over and over. Any soil or used compost can go right back in the bin. Although if you add a lot of soil your bin will cool down.