If matter streams are "fire-and-forget"..wouldn´t they be more like "they are either completely there or they are NOT there"?
Well, it's a "stream" rather than a "package", suggesting that it takes some time to get the entire thing underway. Once that's done, it can survive on its own, and apparently will "bob up" back to realspace at a set distance in the direction it was sent, unless being told otherwise by another piece of machinery. That is, a receiving transporter can take the entire stream or snippets thereof and, among other things, send it back.
"Boosting of signal" is more or less appropriate jargon in this context, I guess: with a weak signal, the stream being sent might "fade", might fail to contain all the bits, just like a wireless datastream or analog audio signal or whatever today. Perhaps some separate "carrier wave" is required to keep the matter stream going from A to B even after it has left A? In that case, the signal (the phased matter stream) could be boosted by ramping up this "carrier wave" even if all of the signal has left A and much of it is already inside B.
But with "Dramatis Personae" in mind, it sounds more likely that this boosting only happens when the stream is inside the machinery, as it survives on its own outside the machinery. In which case it seems some of the stream was inside the SF HQ machine and Kirk wanted all of it safely delivered to the Enterprise
machine, until realizing this was never gonna happen and instead attempting the reverse, that is, pushing everything the Enterprise
already had back to SF HQ. There'd be very little of the signal "in between" A and B, with just a split-second time delay between A and B even if we assume transporters to be limited to lightspeed.