Very fair questions all around. As it mentions in the chapter, the Memory system is an abstraction that models a particular "feel" (any equipment system is to some degree), and what this one is best at modeling is the cinematic dynamic of a character who goes into a situation (mostly the Score) and reveals the various pieces of equipment they brought along with them to the "viewers" right at the instant they are needed. So Ziggy has the right idea: the system generally assumes there is a plan (revealed to us at the table retroactively) around how your user has allocated their memory slots prior to breaking in the door on a Score, and that most of your Memory slots are previously accounted for: whether in equipment we haven't yet seen or, if your User had to relieve stress recently, in treasured high-fidelity memories they haven't yet found the time to unload in a safe location. So what about things that were not part of the loadout "plan" that my user had in mind when they suited up this cycle?
Here's my approach. If the player's user has empty Memory slots they haven't used yet, then this is easy: they can mark those slots with the thing they've acquired in the middle of the score, and we will just say that this was always part of the plan! Having some spare room to stash stuff on the run was in the cards all along for this Score. So that slot is now filled like normal, we're still seeing "the plan" unfold, it just happens to be with things they picked up in the moment instead of things they brought along with them.
BUT, if the player doesn't have empty Memory slots and they really want to make room, I would also let them throw away things to pick up other things on the fly. In this case the slots marked on their Memory track don't go up, but they would cross out existing pieces of Gear and write in new entries in a blank space to make up the balance. For example: "I tossed my Intrusion Kit so I could snatch this guy's encrypted Coinpurse."
Basically as the GM your discretion trumps the rules anytime it needs to. You should absolutely let players do this from time to time, using your discretion as to how this affects the fiction at the table. If it feels like your Memory system is getting abused, you can always start to ask tough questions of the players!
"Hey so... you're actually cool with dropping your Intrusion Kit and leaving it around here? Doesn't that kit have all kinds of identifying information and incriminating details from you on it's timeline?" If they insist on leaving their stuff laying around all the time anyway, well... at least they can't say you didn't warn them when that starts to come back to bite them! One simple and easy consequence is to just assess gear left behind in the Score as an extra point of Heat or two during the Payoff segment. Alternatively, you can get even more devious with it. Does an enemy faction get hold of your stuff? Does this let them start a Long-Term Clock to break the code on your crew's ways and means? Will this, on a long-enough timescale, result in your security protocols getting owned by malefactors, or sold out on the information exchange? Gear integrity can be serious business in World!