Appointing someone else to take over as imam during the prayer

Firstly: 

If a reason arises which prevents the imam from completing
the prayer or from continuing as the imam, he may appoint someone to take
over for him and complete the prayer with the congregation. This is the view
of the majority of te scholars. 

Al-Nawawi (may Allaah have mercy on him) said in
al-Majmoo’ (4/142): 

Ibn al-Mundhir narrated it from ‘Umar ibn al-Khattaab, ‘Ali,
‘Ilqimah, ‘Ata’, al-Hasan al-Basri, al-Nakha’i, al-Thawri, Maalik, ashaab
al-ra’i and Ahmad, and he did not narrate from anyone that it is not
permitted to appoint someone else to take over. End quote. 

They quoted as evidence the fact that when ‘Umar ibn
al-Khattaab (may Allaah be pleased with him) was stabbed as he was praying,
he took the hand of ‘Abd al-Rahmaan ibn ‘Awf and made him move forward, and
he led the people in the remainder of the prayer. Narrated by al-Bukhaari,
3700. 

That was done in the presence of the Sahaabah and others, and
no one objected, so there was consensus. 

See: al-Mawsoo’ah al-Fiqhiyyah (3/252) 

Secondly: 

If the imam becomes unable to complete the prayer or to do
any of the essential parts of the prayer, that is regarded as one of the
excuses which make it permissible for him to appoint someone else to
complete the congregational prayer. 

Abu’l-Waleed al-Baaji said in al-Muntaqa Sharh al-Muwatta’
(1/290): 

With regard to the imam moving back because of an excuse,
there is no difference of opinion concerning the fact that this is
permissible. The excuses are of two types: 

That includes something that makes the imam obliged to become
a member of the congregation, which is if he becomes unable to do any of the
obligatory parts of the prayer. In that case he should move back and make
another man move forward to complete the prayer and he should follow the new
imam. End quote. 

Another example is if the imam is overcome with weeping and
he cannot complete the recitation of al-Faatihah, which is an essential part
of the prayer. In that case he should appoint a member of the congregation
to take over.  

Ibn Qudaamah said in al-Mughni (1/944): 

If he is unable to complete al-Faatihah, then he may appoint
someone to take over leading them in prayer, because he has become unable to
do it. And it is permissible to appoint someone else, such as if he breaks
his wudoo’ by accident or if he becomes unable during the prayer to do an
essential part of the prayer, then he may appoint someone to complete the
prayer. End quote.  

But if the imam has finished reciting al-Faatihah, then he is
overcome with weeping whilst reciting Qur’aan after al-Faatihah, in that
case he may stop reciting and bow, and he does not have to do anything
else. 

It was narrated that ‘Abd-Allaah ibn al-Saa’ib (may Allaah be
pleased with him) said: The Messenger of Allaah (peace and blessings of
Allaah be upon him) led us in praying Fajr in Makkah, and he started to
recite Soorat al-Mu’minoon, until he came to the passage that mentions Moosa
and Haroon, when the Prophet (peace and blessings of Allaah be upon
him) was overcome by coughing, so he bowed. Narrated by Muslim (455). 

Al-Nawawi (may Allaah have mercy on him) said in Sharh
Muslim (4/177): 

This hadeeth shows that it is permissible to stop reciting
and it is not makrooh to do so, if one stops for a reason. End quote. 

It is not permissible for him to appoint someone else to take
over in that case, because he is able to complete the prayer with the
people, doing only the essential parts of the prayer. There is no need for
him to continue reciting with difficulty when weeping, because reciting
something else in addition to al-Faatihah is not obligatory. 

In Mawaahib al-Jaleel (2/136) it is narrated that
al-Maaziri said: 

He should not appoint someone to take over if he is prevented
from reciting part of the soorah. 

In such cases it is recommended that he try to control
himself as much as he can, so that he will not cause any hardship to the
other worshippers if his weeping goes on for too long, as that will have
some negative effects.  

Shaykh Ibn Baaz (may Allaah have mercy on him) was asked in
Majmoo’ al-Fataawa (11/342): What is your opinion on the phenomenon
of weeping loudly? 

He replied: 

I have advised many of those who have gotten in touch with me
to avoid doing this and that it is not appropriate, because it annoys people
and causes hardship to them, and it distracts other worshippers and those
who are reading Qur’aan. What the believer should do is strive not to let
his voice be heard if he is weeping, and he should beware of showing off,
for the shaytaan may lead him to show off. He should not annoy anyone with
his voice, or disturb them. But it is known that some people do not do that
by choice, rather they get overwhelmed without meaning to. In this case it
is forgivable if he cannot help it. It is proven that when the Prophet
(peace and blessings of Allaah be upon him) recited Qur’aan, his chest would
heave like the boiling of a cauldron because of weeping. And it is narrated
that when Abu Bakr (may Allaah be pleased with him) read Qur’aan, he could
not make the people hear it because of weeping. And it is narrated that
‘Umar’s sobs could be heard from behind the rows (of worshippers). But this
does not mean that the voice should be raised in weeping, rather it is
something that overwhelmed them because of their fear of Allaah. So if a
person is overcome with weeping and cannot help it, there is no sin on him
for that. End quote. 

Thirdly: 

From the above it is clear that if your imam had finished
reciting al-Faatihah, then he made a mistake when he appointed someone else
to take over, and he should not do that again. But the ruling on your prayer
and his is that the prayer is valid, in sha Allaah, and no one should be
told to repeat it, because you were all unaware of this ruling.  

But if the imam is overcome with weeping whilst reciting
al-Faatihah, then what he did by appointing someone else to take over was
correct. 

And Allaah knows best.