In JO, the exact rule is
"After a fall from the bars, judging resumes with the performance of an element that is listed in the Jr. Olympic Code of Points."
So, if the gymnast gets back to the high bar via a low bar kip followed by a squat on, yes, the kip and the squat on are judged. This matters in L10 where only one squat on is allowed per routine. If the gymnast jumps or is lifted to the high bar (with a push from the coach allowed), they get two unjudged pump swings to get momentum, then judging starts so any extra swings past two get an empty swing deduction.