What is the ruling on the prayer of one who bowed twice in a single rak'ah inadvertently due to the movement of worshipers during the prayer, and what are the consequences for the ma'mum (follower) who caught up with one of the rak'ahs after the repeated bowing?

If the Imam mistakenly performs an extra bowing (ruku'), and a latecomer (masbuq) joins him in this extra ruku', then the latecomer is not considered to have caught the rak'ah, and he must complete a full rak'ah after the Imam's salam. This is because this extra ruku' is considered null and void (laghw). This is the view of Ash-Shafi'i, An-Nawawi, Ibn Nujaym, and Al-Qarafi, and it is what the four madhhabs are upon. It is not analogous to catching the extra fifth rak'ah.