I used to get paid for writing on this kind of thing and I'm well enough read and I've talked to scores of pilots. Always room for error though. As noted there's no complete agreement on planes. I'll stand correction on this matter - but here's how the jet stacks up as I understand it.
All Things Being Equal:
1. The jet was much faster than a P-51. It was also heavier and could lose energy faster. However, it had two alternate tactics in a one vs one (or 2vs 2 or 3vs 5). One was simple boom and zoom: 30mm canon are fierce and no 51 would trade head-ons intentionally. A pony could dive out, but if there was enough alt, the jet could track it down. Best outcome is 51 gets away.
2. This didn't make much sense to me at first, but I think I get it now. The 262 actually handled very well and had a tight radius for its weight and speed. It couldn't turn inside a P51 but if the speed advantage was near max (say +90mph) it could turn outside the Mustang and give it a burst - one round hits and the 51 goes down. (In other words the 262 is going around a larger circle than the 51 is going around a smaller one.) The danger to the jet pilot would have been extending the turn too long and losing too much E - the jet took a while to regain E in level flight and then advantage goes to 51. (Very like a P-38 or P-47 vs a Zero.)
3. The LW thought the jet was very much a fighter - Hitler had an odd point about the "blitz bomber" because land morale declined seeing nothing but enemy. But its designers and all responsible LW leaders saw fighter, and that's how most functioned. The reason you rarely heard about dogfights between jets and fighters was because jets were ordered not to engage in them. Killing fighters wasn't their job, and it was bad to try it. That said, jets bagged several allied fighters. 51 pilots had a tactic called "boxing" where four aircraft would attack at an angle that should give at least one of them a shot if the jet had slowed down for any reason. If it didn't the jet was simply to fast to fight. So P51s had a good shot if they outnumbered a jet 5-1 - and that was the norm. (Jet pilots were not experts by and large - they were told to keep it simple.)
4. The 262 met its Waterloo when USAAF and RAF fighters started standing patrols over every base that could handle a jet. Several were downed when landing or taking off. Indeed, most allied victories came in one stage of that or another. The other thing that could happen was that a 262 pilot would lose concentration - they only went full throttle during an engagement. So a flight of Mustangs could attack unseen ... you can guess. (Adolf Galland suffered that fate. He was badly wounded by a 51 and ended the war in a hospital - he believed the 51 saved his life.)
5. Check Korean War tactics and it's very clear that great speed could make a well flown jet almost invincible against a piston fighter. The 262 was a remarkable plane. The Mig15 and F86 airframes both derived from it.
Eric