Will badges I obtain reappear in the panel?

I just started the game for the first time today. I did the tutorial and went to the Practice Catcher, where I won some free plays. I took them over to the Badge Catchers and found a badge that I really really want! Sadly, it was buried under a bunch of other badges. I was able to collect enough badges to clear a path to it, but it took all my plays to do.

I'm pretty sure that duplicate badges can be earned because the tutorial showed off a home screen with many SMB1 coins on it. Additionally, there's a feature to show you which badges you've earned in any given panel, which would be useless if earning a badge removed it forever. So my assumption is that all those badges I collected will be put back into the panel at some point.

Is this correct? If so, when does it happen? If the badges don't come back, I'll try and earn the badge I want for free (but I'll spend money if I see that it says "Final Day!"). If they are going to come back before I get a chance to earn more free plays, I might as well spend my money now while I feel good about my chances.