I don't know if this helps, but the escort distribution has a simple interpretation: you have q replicas of the system, and you are selecting to view only those instances when all q replicas have exactly the same state. This situation occurs with probability $p_i^q$, so the formula follows.
I can only imagine highly contrived situations where such a thing would be physically relevant. Perhaps you are looking at q atoms with k levels, which fluctuate statistically, but only when they are in the same level can you get electron flow through the system, because only then do you have resonant mixing of the states. Then if you restrict attention only to those cases where the electron manages to hop through the system, the average of the physical quantities of the atoms in those trials will be governed by the distribution you give.
Unfortunately, there seems to be a large literature on this, and I have no idea what physical systems it describes.