If a research group downloads material in order to train a model, is there some significant difference in copyright violation if they hand it to a second research group in order to fulfill the same purposes?

Yes, because of a key word in a lot of copyright laws... "distribution". Using that copyrighted material themselves to train the model still gives them plausible deniability. Handing the copyrighted material to another group starts to run afoul of other laws and also removes the plausible deniability that the original group can claim regarding their training data.

The training data is not necessarily kept. It's possible that data is consumed, incorporated into the weights and then discarded.

If only we'd figured out a technology that let us move huge torrents of bits around.

If only there was a catchy name for it. Something like bit-torrent perhaps?