I used to think that A-theorists cannot consistently believe in time travel. I think I was mistaken. As best as I can reconstruct my line of thought it was this. Time travel requires a distinction between external and internal time. If I go into a time machine, then maybe in five minutes I'll be a thousand years ago. That's a contradiction given non-circular time unless one distinguishes as follows: internally in five minutes I'll be a thousand years ago externally. But now I think that what I must have been thinking was that in five internal minutes my internal present will no longer line up with the world's objective present, since in five internal minutes my internal present will be about thousand years behind the world's external present. I don't know for sure if that's the thought I had, but if it was, it would have been a howler. For on the view, internally in five minutes, I will be at the time at which the world's external present was about a thousand years ago. Or, to put it from the external point of view, a thousand years ago I was five minutes older than I am now (age is measured internally). Even presentists can say that.
To see that this is coherent, consider a theory that takes external time to governed by the A-theory but internal time to be entirely governed by the B-theory. Thus, superimposed on the external A-series of past, present and future, there is an indexical B-series of earlier-for-me and later-for-me, where these relations are perhaps defined by internal causal relations (earlier states causing later ones). There is no more need for these two series to line up than there would be a need for the two series to line up if the external series were a B-series.
However, while this is coherent, maybe it undercuts one of the main motivations for the A-theory. For if there is a distinction between internal and external time, as there must be for time travel to be possible, all the changes we actually experience are changes with respect to internal time. In other words, they are B-type changes. But the typical A-theorist thinks B-type changes--it (internally) earlier being one way, and (internally) later another--are not what we experience when we experience "real change". Indeed, if time travel is possible, it is possible to live all of one's life at one external time, but moving through external space. Basically, just imagine that at each moment you travel to some external time t0, but to a different spatial location in it. Maybe you have a backpack time-machine which is permanently stuck on t0, but with the spatial locations changing. You'd experience change, because your state at internally earlier times will be different from your state at internally later times. But it would be mere B-type change, since it would all be happening at one and the same objective time.
I suppose one could say that in time-travel scenarios, especially the preceding one of living all of one's life at one external time, our experiences of change become non-veridical, for a condition on the veridicality of our experiences of change is that our internal clock lines up correctly with external time, and time-travel causes a misalignment. Maybe.
But in any case, now that we have the possibility of living all of one's life--a life that presumably could have rich causal interconnections--at one objective external time, just moving "sideways" to new spatial locations, I do think that the motivations for the A-theory decrease. For we see that what matters for the diachronic richness of our lives is that our lives be stretched over internal time, not over external time. It also matters that other people's internal times be sufficiently lined up with ours. But that doesn't call for the A-theory, either.
So, all in all, while A-theorists can believe in time travel, thinking time travel through would undercut much of the motivation for the A-theory.