Getting the waste to a storage facility is very easy. You put it on a semi trailer and move it to the storage site: http://large.stanford.edu/courses/2015/ph241/avery-w2/images...

Nuclear waste is radioactive, but not so radioactive that it's unsafe to stand next to a container. Most waste is stored on site.

It actually doesn't make sense to move nuclear waste to permanent storage because some reactor designs can use this waste as fuel.

It’s not really much of a problem. You leave the waste in a cooling pond at the plant for a couple of decades while you wait for all the really threatening stuff to decay, and then drive it where it needs to go in a truck. It’s a bunch of big metal rods in canisters. It can’t really “spill” and if it does you just pick it up and put it back in the truck. Uranium and plutonium are really not very threatening to human life.

Besides the obvious NIMBY problems, are there any problems with sealed casks transported by train?

No. These things are tested to a VERY high standard.

https://www.youtube.com/watch?v=1mHtOW-OBO4