Denis, as it was, the DPD were on the scene in record time. Same goes for converging on Oswald in the theater. But they had a deal with the FBI that they would not be responsible for offing Oswald. They would only deliver him so they could claim plausible deniability. Their handling of the crime scene was incriminating enough. Killing Oswald would have opened a whole new fresh can-o-worms.
At some point Oswald suspected a double-cross was afoot and made a break for it, perhaps ahead of schedule. He had no choice but to head towards the post-coup pick up point where his handler would either help him escape, or kill him. By this point, his best bet was to surrender to the DPD in a public setting. Oswald was safe with them, so he thought, until they delivered him to Ruby on a silver platter.
WOW!!....You've summed it up very nicely Mr Trojan..... In A nut shell. I'd like to add a little "food for thought" in a rhetorical question.....
What the hell were THREE FBI agents doing in the Texas Theater at the time the cops burst in to arrest a man who had allegedly sneaked into the theater without buying a ticket.

?