Are you a DVC member, or are you renting points?
If not a DVC member, are you an AP holder?
If you're not staying on points and/or you're not an AP holder, staying at 3 different resorts and buying the DIsney DIning Plan at each is a very unwise financial move, since you'll have to buy at least a 1-day park ticket for each resort stay.
But let's say that doesn't apply to you ...
If you want the Quick Service Dining Plan during your entire stay, then you will need to book it separately for each resort stay. You will get a mug per person for each of your 3 stays.
But if you just want the QSDP for one stay, and are concerned about mug use at the other 2 resorts, don't be concerned. The mug rules no longer state that the mug can be (legitimately) used only at the resort where it was purchased. The mug rules now state that it must be used during the stay in which it was purchased (it said this part before, also). You are doing one "stay" at Disney resorts, just across three different resorts. Following the rules, you can use the mug from the first resort at all 3 resorts during your 9-day stay at Walt Disney World resorts.
Signed, The Mug Police