If a county official is sued in his or her official capacity, and the case ultimately results in an injunction against the named official in his or her capacity, does that injunction bind the official's successor in office if the official leaves office before the injunction expires? Can you sue a public office without necessarily suing the person occupying that office at the moment?

This is why people don't pay as much attention to the civil forum. A relatively simple question can result in a painfully obtuse answer.

If the suit was in federal court, your answer is easy: it's "yes" to both questions. See Fed. R. Civ. P. 25(d)(1), (2).

There's no such pat answer in Texas practice ... at least until you reach the appellate level. See Tex. R. App. P. 7.2(a). Since appellate Rule 7.2 essentially tracks the federal rule, an identity of purpose and intended effect can be seen. Thus, it could be argued (persuasively, I think) that the same principles apply in the trial court. If they don't, in fact, then why is the appellate rule what it is?

All of this bolsters the foundational rationale for why the answer to your question should be "yes"; an official capacity suit is one against the office served by the defendant, not the defendant himself/herself.