1. If you enter the masjid and find an individual praying:
a. Do not ask him what Salāt he's praying (You can join anyone in Salāt irrespective of Salāt).
b. Stand next to him to his right.
c. You don't need to do iqāmah, just make Takbīr.
a. Do not push or pull anyone. Stand to the right side of the follower of the Imām. The Imām either takes a few steps forward, or the follower takes a few steps backward together with you).
a. Do not touch him to notify him of your presence
b. Form a row behind him & make your takbīrat loud enough for him to hear.
4. If you find the individual or congregation on the final tashahhud
b. After their taslīm, rise to complete your Salāt individually, or, preferably, start your own Jamā'ah with a fresh iqāmah.
5. If you enter the masjid and find someone who rises to complete his Salāt after the Imām's taslīm,
I hope this helps.
Sabāhul khayr