No explanation needed, can a P-51 Mustang outrun a missile?
The P-51 Mustang is not designed to go supersonic, in fact no propeller airplane is designed to do that...
For example, the Republic XF-84H is the fastest propeller plane ever built, is also the loudest. The propeller tips going supersonic mean that the airplane is incredibly loud, so loud that it causes injuries.
Anyway, the Thunderscreech can hit top speeds of about Mach 0.9. A stock P-51 Mustang can hit about 440mph. Missles, like the AIM-9 Sidewinder can travel at speeds exceeding Mach 2.5. Mach 1 at sea level is around 760mph. Mach 2.5 is 1,918mph. The AIM-9's range is 22 miles, so it can travel 32 miles per minute or about half a mile per second. At the Mustangs top speed of 440mph, the only way the Mustang could "out run" the missile is if the Mustang was about 15 miles ahead traveling at top speed. Even the Thunderscreech would have a tough time outrunning one at full speed without a significant head start.
So the short answer is no, the Mustang is not capable of outrunning an air-to-air missile. Even surface-to-air missiles have a speed between Mach 1.75 and Mach 2 with similar ranges.
Early missles, like the K-13 built in 1960 had a top speed of Mach 2.5, so it wouldn't matter if it happened 50 years ago or today.
In counterpoint to @Ron Bayer's answer:
During the V-1 attacks in 1944 and 1945, modified P-51s (among other aircraft) were used to intercept and destroy the "cruise missiles" before they reached their intended targets.
Both gunfire and "tipping"(putting your wing 6 inches under the V-1 wing and using the airflow to tip the V-1 over) were used. P-51 Mustangs destroyed 232 V-1s.
Technically, the P-51s weren't "outrunning"; they were catching...
I would just like to add that considering most early missiles were heat-seeking and infrared guided, the P-51's heat signature thanks to its relatively cool running inline wouldn't be a sure detection by the seeker head. However, the all metal, HIGHLY conventional design would be an easy pick for radar-seeking seekers. If that adds anything to your question.