What should one do if they miss part of the prayer for rain (Istisqa) or Eid prayer, such as someone who catches the second unit (rak'ah) or misses the bowing (ruku') or prostration (sujud) in it?

The correct opinion among scholars is that what a latecomer (masbuq) catches with the Imam is the beginning of his prayer, and what he completes individually is the end of his prayer. This is based on the saying of the Prophet (peace be upon him): "So whatever you catch, pray, and whatever you miss, complete." There is no difference in this between the obligatory prayers, the Eid prayer, the الاستسقاء (prayer for rain), and others. So, what the latecomer catches is considered the beginning of his prayer, and he completes what he missed based on that. Thus, he performs the takbir (Allahu Akbar) in the subsequent rak'ahs (units of prayer) according to their position in his own prayer, not the Imam's prayer.