It was definitely not working correctly for me. And the issue was not that it took too long to update.
I played one game on the new patch since the new missions came out and I had 3 units with 3 items by the time the game ended (you can check my match history or I can provide a screenshot for that if you want proof).
I have played a second game a few hours later.
After the first game, it did update... but only one out of the three units actually counted towards mission progress.
Maybe it was an issue with the specific galaxy (Treasure Trove) or the fact that some of those items came from the Kayn boss?
Thank you for answering though!