If he catches up with the imam when he is prostrating or sitting

Firstly: 

If a worshipper enters the mosque when the imam is
prostrating or sitting or whatever, he should join him, because the Prophet
(peace and blessings of Allaah be upon him) said: “If you come to
prayer and we are prostrating, then prostrate but do not count that as
anything (i.e., as a rak’ah), and whoever catches up with the bowing has
caught up with the prayer.” Narrated by Abu Dawood, 893; classed as hasan by
al-Albaani in Saheeh Abi Dawood. 

See also question no. [46811]. 

But this rak’ah does not count for him, because he did not
catch up with bowing with the imam. 

Secondly: 

The scholars (may Allaah
have mercy on them) have stated that the worshipper who catches up with his
imam when he is prostrating should say the opening takbeer and then
prostrate with his imam, without saying takbeer, because he has not caught
up at a point where takbeer is required. 

Ibn Qudaamah said in al-Mughni (2/183): If he catches
up with the imam in a pillar (obligatory part of the prayer) other than
bowing, he should not say any takbeer apart from the opening takbeer, and he
should go down (in prostration) without saying takbeer, because this
(rak’ah) is not counted, and he has missed the point where takbeer is
required. If he catches up with him when he is prostrating or in the first
tashahhud, he should say takbeer when he stands up with the imam for the
third rak’ah, because he is following him, so he should follow him in saying
takbeer, like one who catches up with him at the beginning. 

See al-Majmoo’ Sharh al-Muhadhdhab, 4/218 

This is unlike the
takbeer for bowing, which is counted for him, and it is unlike what he
should do after that when he moves with the imam, getting up from
prostration etc. He should say takbeer, following the imam in his movements,
even if it is not counted for him, as stated by Ibn Qudaamah (may Allaah
have mercy on him) 

The second opinion is that he should go down in prostration
with the imam, saying takbeer. So he should say takbeer the first time to
enter the prayer (opening takbeer), and the second time to prostrate or sit,
because he has to follow the imam, who is sitting or prostrating, and when
one moves from standing to prostrating, there should be a takbeer. 

See al-Majmoo’, 4/218; Haashiyat Ibn Qaasim,
2/277; al-Insaaf, 2/225; al-Sunan al-Kubra, by al-Bayhaqi,
2/91 

Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said in
al-Sharh al-Mumti’, 4/128: 

The well known view of
the fuqaha’ (may Allaah have mercy on them) is that he should go down (in
prostration) without saying takbeer. 

But despite that we say that if a person says takbeer it does
not matter, and if he does not say it, it does not matter, and we say that
it is the person’s choice, because there is no clear evidence to
differentiate between bowing and anything else. So it is possible to say
that sitting does not come after standing, but what made me sit is the fact
that I am following the imam, so now I am moving to an essential part of the
prayer that I am enjoined to move to. But I am doing that to follow the
imam, not because this is how the prayer should be in principle. Undoubtedly
he supports the view that he should say takbeer, so what we think concerning
this matter is that in order to be on the safe side he should say takbeer.
End quote. 

Thirdly: 

If he enters the prayer and goes down in prostration, then
the imam raises his head before he puts his forehead on the ground, then it
seems that he should come back up with him and not prostrate, because the
time for following him (in prostration) ceased when the imam raised his head
from the ground before the person put his forehead on the ground, unlike if
he had been with him from the beginning of the prayer. 

If he catches up with
him in the first prostration and he goes down in prostration then the imam
raises his head, and sits between the two prostrations, the person should
sit with him, then when he does the second prostration he should prostrate
with him. 

And Allaah knows best. 

See Ahkaam Hudoor al-Masaajid, p. 143-144, by Shaykh
‘Abd-Allaah ibn Saalih al-Fawzaan.