I'm sure it doesn't if you do it right, not entirely sure how you are doing it wrong thou.
I have 3 sets, one based on crit (used for fox/coin trial), two based on init with one of them having the highest init on main. They never get messed up unless I do something wrong, like accidentally remove 2 or more ninja from the formation.

try to remove them 1-1 after you remove 1 ninja put the one you want instead of him... so basically lets make an example... 1st position you have chojuro with your best equipment and you want to put sage naruto in 1st potition with his best gear.. so you remove chojuro and put sage naruto immediately so your gear will not messed up.. and continue wit the same way to your other ninjas aswell

Alternatively, you can drag and drop the ninja you want to put IN over the one you want to put OUT.

That's extremely odd. As gear stays in place depending on the Set you've placed.

I'll run a test later today to confirm if it's moving around.