How would that even work?

You pass out cards that get signatures. If there are enough signatures, the teacher's union, which is AFL, swoops in and says "dear University, we have a preponderance of support from your grad students to negotiate on their behalf so under the labor laws of this state and nation, you get to talk to us." From that point forth, any grad student who teaches at that union has to pay the union dues. In exchange, the union bargains for their wages and benefits. It's a fairly simple process, actually.

Grad students are generally in a competitive and underpaid market

That "underpaid" thing is why you unionize. The union says "yeah, so we can make this really ugly unless you start paying these students a living wage."

aren't in that market for a significant amount of time

The union says "we're going to take your benefits contributions and put them in this handy 401(k) that's portable to wherever you end up next." next question?

and for many of them are trying to become the employers that they'd be collectively bargaining against.

Really? They're going to become state legislators? Well then they won't really care much because generally state workers have pretty sweet pensions. Not only that, the money isn't coming out of their pockets it's coming out of their constituency.