I see bug reports on this subject, so apparently some of y'all understand how this is supposed to work. I'm afraid I don't.
If you want to know what I've tried that doesn't seem to work, keep reading. If you prefer just to give step-by-step instructions, you can ignore what is undoubtedly mostly noise w/ very little signal below.
When I choose Apply at the smithy, I get the upgrades I've so far bought in the left-hand grid and the army members in the town on the right. Fine. But then what? Unlike other similar grids, there are no instructions at the top of the display!
At one point, with Type and Upgrade selected, I tried pulling an upgrade onto the right-hand grid on top of my 5 Legionaries. Nothing indicated what had happened, but when I chose Type only at the bottom of the page, they separated into 3 and 2!
Right-clicking soldiers in the right-hand grid doesn't move them visibly to the left, as it does when selecting an Attack or Reinforce expedition; they just disappear. Selecting and dragging them may work, for all I know. But when I choose OK, I can't find anything that shows different when pointing at a soldier. And when I choose Apply again, the upgrades are still there, so I suspect they haven't been applied.
What am I missing that's been obvious to everybody else???