Is there a free software that can do this? So far I have looked at paint.net and GIMP, both of which unfortunately do not support this feature. All help is appreciated!
One possibility is to use vector software. Inkscape is free and Open Source.
In the example below I have a bunch of blue squares on one layer, and a bunch of pink squares on a layer above. You can see me toggle the layers in the example
Use the Select and Transform Objects tool F1 to make a selection around the squares you want to move, and it will also select the hidden squares in the layer below. Holding down Shift adds to the selection when using this tool.
Then you can move all those selected as once, the pink and blue squares moving as one, even though they are on different layers.