Most likey they will use the same service, and you will get your dumps back pretty quickly. It is also possible that the bank or coin service may just bring your dump box right over to the pick-up bank. I have had that happen to me before where Loomis delivered to my pick-up bank a brinks box full of searched halves that someone had dumped at another bank on their route.
Even if that doesn't happen, I would still strongly recommend against dumping at the same Bank where you pick up. Tellers do move between branches, and do talk. If they find out you are picking up at one bank and dumping at another, they may be more likely to cut you off.
If you are going to have banks order boxes, my advice is to open accounts at several banks, and then use some as dump banks and others as pickup banks. If you are just searching rolls that they have on hand, then it probably won't be an issue.