Main reason for checking this thread was to see of my cynical belief held true- that the question must have been posted by someone with a very low post count.

Not that there's anything wrong with that. We all had to start sometime!

To help with the OP's question, the only time it would be relevant to ask would be the same time it would make most sense to hoof it as quickly as possible to the gate, rather than spend any time trying to ask a gate agent to do something that in all likelihood will be a waste of time. The new United, the one reborn out of bankruptcy that tried to be first at something, chose on-time departure. And by gummit, they accomplished it. Something they could trumpet on paper napkins that they were #1 at.

By doing so they changed the... culture?... of the gate agent/passenger interaction. When it's time to fly, it's time to fly! And that time is based upon scheduled departure time, not the expected arrival of passengers from a late flight. Cruel? Yes and no. Holding up flights can cause ripples down the line, unintended circumstances that delay other passengers who then miss their flights etc.

Having said all that, the gate agents are aware of late-arriving passengers and they may, in some cases, wait a short amount of time for them. I've seen this happen once in a while, although not nearly as often as in the past. And no, it doesn't seem to make much difference if it's the last flight out of town or not.

But the short answer is, RUN! Don't talk.