I don't really love this because it seems wasteful to DFA and trade him instead of optioning Francis when they've been pretty equally bad.
The only explanation I can think of is that Francis is younger and this season has had (in their limited appearances) better strikeout numbers and has been more impacted by BABIP than White, both of which would probably tend to bode better for a rebound, whereas the pro-White side of things is largely "he was really good for the back half of last season in Buffalo) and that could be a bit of a mirage that only serves to demonstrate that he's a AAAA type of pitcher at best.
Or that they had some sort of agreement with White where he would get his shot to start this season but if it didn't work they'd let him move on rather than stick him back in the minors. DFA-and-trade would be an easy route for that because it would end one of 3 ways: 1) he gets claimed by another team who gives him a shot, 2) he elects free agency after clearing waivers and gets to sign elsewhere, or 3) seeing that (1) didn't happen and (2) looks likely, the team trades him somewhere just to get anything in return rather than lose him for nothing.
And I guess the biggest thing is that fretting over which of the two underperforming arms gets the role of lowest guy in the bullpen totem pole is kind of pointless.
