Barrels/pots on a stockpile tile get assigned to that stockpile tile and tend to keep that assignment when moved elsewhere.
This is helpful when you want your alcohol in one place, but not so great when you're using minecarts or water to QSP it.
A low level workaround is for your barrelcarts to not sit on track stops and dump them straight out, instead having to wait till full (25 barrels) at least, deleting and recreating the input stockpile by then.
The tile the cart sits on can also be part of stockpile, of course (even though carts guard against rot by themselves), though be warned, anything in a cart when it is assigned on a route tends to be inaccessible for jobs, an useful safety measure (otherwise, there's no problem storing barrels in carts and then having that cart sit on a stockpile - here, let me just remove that cart from route and assign furniture/booze stockpile beneath it...Linking it and both food stockpile with single seed to kitchen)
And yup, easy boozecooking - and the barrel will only now be moved back to the stockpile minecart took it from.
PS: I think dwarves can drink from the barrels in that cart, but only one at time - watching over a bit of time sees drink stock drift downwards but also some dwarves going to drink from elsewhere, because they drink in waves. Consider using waterskins anyway.
(A high level workaround would be to make booze production and booze qsp unpathable for dwarves from each other.)