What would be a single word for the process in which political parties (or other groups that have something to gain) embrace and extend (and maybe corrupt) the ideas of a political movement that started and grew outside of the regular channels? Recuperate?
Perhaps you want the word co-opt.
a : to take into a group (as a faction, movement, or culture) : absorb, assimilate. the students are co–opted by a system they serve even in their struggle against it — A. C. Danto
b : take over, appropriate. a style co–opted by advertisers
Appropriate (as a verb, pronounced with a long final a, to rhyme with state) may also serve your purpose.
Recuperate is almost always used to mean "to recover from an illness". Its alternate meaning of "to recover something that was lost" doesn't fit here either. Merriam-Webster also gives "to bring back into use or currency : revive" but that seems very uncommon, and is no better.