Not foam core, but if you clear coat the minis, I don't see how the minimal amount of movement in the tacklebox is gonna cause any problems. This solution holds all released minis and more when some of the slots are doubled up. I considered foam core, but with all the minis it seems to be a large waste of space.

For minis: Plano 1374 4-By Rack System 3700 Size Tackle Box https://www.amazon.com/dp/B000E3FKTO/ref=cm_sw_r_cp_apa_i_Bk8qDbWMSSYHP

In use: https://imgur.com/gallery/WIdy2A9

For tiles : https://www.reddit.com/r/ImperialAssaultTMG/comments/a9v2t1/imperial_assault_tile_storage_solution

I went with the Plano 137401. I couldn't find one which comes with the 3750s, which I've seen are better for my minis since the larger slots hold ogres, giants, etc. and there are more small slots which are perfect for medium and small humanoids. Fortunately I already a couple of 3750s and can mix-and-match for now. :)

Here you go. https://www.amazon.com/gp/aw/d/B000E3FKTO/ref=cm_cr_othr_mb_bdcrb_top?ie=UTF8#

This tackle box is $50. I haven't bought it for my Firefly game yet. But, I have the same concern you do and this looks like a fairly inexpensive solution. If you look at the reviews several people have used it for similar types of games. Hope this helps.