It might be easier to see what it looks like from the fiber functor perspective. The torsor of etale paths = Isom(F_x, F_y) for the fiber functors F_x, F_y. An iso. is essentially given by maps between the fibers Y_x and Y_y for all covers Y. Given such a map you can act by Gal on both sides and get a new isom. If I didn't make a dumb mistake that's basically the Galois action. Then you can always translate back from fiber functor picture to check what this action corresponds to.

Is strictness really necessary? I seem to have found a source which suggests henselianity suffices (see Theorem 8.1 of mi.fu-berlin.de/users/kindler/documents/basechange.pdf). Of course I should have looked for this before asking the question...but it seems believable to me. He works over an algebraically closed field but comments that it's not necessary...I guess that's the only possible hitch?