Well first, 1U Thermals and noise are terrible due to the need of smallers fan who don't move as much air, but it's probably fine with NUC hardware as they are pretty low power. I would go for 3u or 4u at the very least if I actually want to move them.
Although you could just mount them in their own case.
Why do you need to transfer just the internals? There are options out there that will allow you to rackmount the NUC in its case. Racknex are the best ones but MyElectronics ones might also work.
Haven't done it myself, but I'd say the most important thing is to not short anything out. A lot of 1u cases already come with a plastic sheet on the bottom.
Maybe see if you can get some sort of stand you could tape or screw into the case and then the NUC onto that. Maybe NUC board to itx screw holes adapters exist, don't know.
That's where I'd start, next thing is getting the ports accessible. Most stuff is available as keystones, at least USB and such. If you can 3D print, an adapter for the io shield to hold some keystones shouldn't be too hard to make.
Add some case fans as well and that's about it I'd say.