I asked for a measurement which could identify such a group, not a measurement that would. I can easily tell you experiments to test this in physics - solve Newton's law of motion and find a celestial body with motion that doesn't agree with it.
If I were advocating for the invisible roller coaster track theory of celestial motion, I couldn't provide such an experiment. The invisible roller coaster tracks are observable only by celestial motion - whichever way the moon moves, that's where the track is.
The only way to refute the theory would be via an alternate method of observing the position of the tracks and then observing whether the moon actually followed that track. If someone didn't provide that alternate method, I'd say he was not even wrong.
However, if you have specific questions...
Besides the one I repeatedly ask, you mean?
Examples from the middle ages include grants of knighthood as payment for some unusual service. While usually a knight would only come from wealthy or noble families (or at least a family with good connections) -- hence, from a position of some power -- sometimes knighthood was granted to brave foot soldiers -- i.e. people with little power. Sometimes, the title came with land (and the serfs that worked it, of course).
In non feudal societies, social mobility was usually achieved through money, although some classes were barred from obtaining any money whatsoever (slaves). You can see groups of immigrants, provided the host society did not block their steps too much, slowly gain money, and later recognition and connections. This process would often take several generations.
Analyzing those processes is helped by the fact that often you can observe power directly. Money and nobility titles are very conspicuous forms of power, easily measurable directly. More hidden forms of power such as connections can also be traced directly (a boy of low background would be taken to the home of a merchant as a gift to his parents in recognition of some service; this lets you trace connections across classes); charisma (which in the middle ages was a great way to attain power in religious circles) could be seen in some extraordinary ascetic acts or visions. The latter was one of the few ways women could rise to positions of power in medieval societies (see Joan of Arc), although others would be marrying, and surviving, a man of power. While it was often expected of widows to remarry, some medieval societies were surprisingly relatively accepting widows, recognized their independence, and allowed them to transact on their own.