Do "Current Missions" cards get removed as you play them and move into "Played Missions"?

I'm looking at my list of cards in "Current Missions" and was wondering if the cards get moved over into the "Played Missions" category, as a way to eliminate from the deck.

For example, there are qty-3 Wild Mission Cards, as I play each one, in theory should they not be eliminated from the deck because I played that mission ID#. Eventually, as I play all qty-3 then qty-zero will appear in "Current Missions" and qty-3 will appear in "Played Missions"?

Or, over time, do I just continue to add cards to my "Current Missions" deck so it looks like I have loads of missions to play? What happens if I rack up like 100 mission cards? I'm assuming there will be filtering.