If a cart isn't being recognized or is "missing" from the board's perspective, it will be skipped when you hit Select Game and the next available cart will be selected. I would remove the carts, clean the cart and slot contacts well, and reseat.
On my 2-slot I sometimes have to insert the cart until it is fully seated, then back it out by maybe 1/8" to get it to be happy. Sometimes even adjusting the "tilt" of the cart in the slot does it. Once I have a game booting reliably with no audio or graphical glitches it stays stable but it's sometimes finicky (on mine at least) to get a cart to a happy place when first swapped in. Whether you removed/reinserted the carts before/after transport or you transported it with the carts installed (which might have put stress on the slot connectors), they may no longer be seated right.
I'd also obviously check the switch on that Select button, make sure a contact didn't come lose or the micro switch itself dislodged during transport. But if pressing that attempts to switch games and just restarts the one cart that is booting right, the button/switch is probably OK.
Why didn't my dentist have a NeoGeo when I was a kid? Maybe I would have gone more often…