I know with tokens related to missions when someone is withdrawn, they drop the token where they were killed.

Does the same apply to crate tokens?

I don't have my Campaign Book in front of me so can't look at how things are worded there, but Crate Tokens and Mission Tokes have separate entries in the RRG.

Crate Tokes are claimed, which means put in front of you. Most Mission Tokens to my recollection are kept with the figure and not claimed.

So no, crate tokens should not be dropped if the hero Withdraws.

Of course I missed the entry for "Retrieving Tokens" (RRG pg 22).

So it's definitely a no. Crates are Claimed (placed in front of you), all other tokens are Retrieved (placed under your figure).