There is nothing wrong with leaving the mosque after the adhaan to go and pray in another mosque
Question

                It is clear in the sharee3ah that you are not allowed to leave a masjid before the salaat after the Athaan is called unless you have a valid excuse like needing to leave in order to complete you wuDoo. But is it allowed to leave after the athaan in one masjid in order to go to another masjid for the same prayer ? The other masajid may have a bigger jamaa3 or a better reciter as well as having people he wants to see after the salaat. In my case there is a small masjid that has a lesson after salaatul-maghrib up until `Ishaa time but there are other masajid that have bigger jama3s for `Ishaa some miles away in the same city and sometimes have better reciters and also have many other brothers I like seeing.
Praise be to Allah.It is haraam to leave the 
    mosque after the adhaan without an excuse, because Muslim (655) narrated 
    that Abu’l-Sha’tha’ said: We were sitting in the mosque with Abu Hurayrah 
    when the muezzin gave the adhaan. A man stood up and walked out of the 
    mosque, and Abu Hurayrah followed him with his gaze until he exited the 
    mosque. Abu Hurayrah said: This man has disobeyed Abu’l-Qaasim  (peace 
    and blessings of Allaah be upon him). 
It was also narrated by 
    al-Tirmidhi, who commented: Based on this, according to the scholars among 
    the companions of the Prophet  (peace and blessings of Allaah be upon 
    him) and those who came after them, one should not leave the mosque after 
    the adhaan, except for one who has an excuse, such as not having wudoo or 
    something that cannot be avoided. End quote. 
Ahmad (10946) narrated it 
    with the addition: Then he said: The Messenger of Allaah  (peace and 
    blessings of Allaah be upon him) commanded us: “If you are in the mosque and 
    the call to prayer is given, none of you should leave until he prays.” 
    Classed as saheeh by Shu’ayb al-Arna’oot in Tahqeeq al-Musnad. 
Excuses which make it 
    permissible to go out include: going out to do wudoo’ if the place for doing 
    wudoo’ is outside the mosque; going out with the intention of coming back, 
    such as if a person goes out to wake up his family, for example, then come 
    back. This applies to going out to pray in another mosque, if he knows that 
    he will be able to catch up with the prayer there. 
It says in Ghaayat 
    al-Muntaha: It is haraam to leave the mosque after the adhaan and before 
    praying without an excuse or the intention of coming back. … And it is 
    permissible to go out if one goes out after the adhaan but it is in order to 
    pray with the congregation in another mosque, especially if its imam is 
    better. End quote from al-Ghaayah ma’a Mataalib Ooli al-Nuha 
    (1/304). 
Shaykh Muhammad ibn 
    Ibraaheem (may Allaah have mercy on him) said in al-Fataawa (2/104): 
    The prohibition on going out of the mosque is subject to further discussion: 
    if there is no need for it and it serves no valid purpose, then it is haraam 
    to go out. That is because he looks like one who is going out of the mosque 
    without praying. But if he intends to pray in another mosque, or he has an 
    excuse or he intends to return and there is sufficient time, then it is not 
    haraam. End quote. 
Shaykh Ibn ‘Uthaymeen (may 
    Allaah have mercy on him) was asked: What is the ruling on a man doing 
    wudoo’ in one mosque then praying in another mosque? 
He replied: If the place 
    for doing wudoo’ is outside the mosque, there is nothing wrong with him 
    doing wudoo’ in this place and praying in another mosque, unless the iqaamah 
    has already been given in this mosque, in which case it is better for him to 
    pray there. If the place for doing wudoo’ is inside the mosque, then it is 
    not permissible for him to leave the mosque after the adhaan unless it is 
    for a reason, because Abu Hurayrah saw a man leave the mosque after the 
    adhaan and he said: This man has disobeyed Abu’l-Qaasim  (peace and 
    blessings of Allaah be upon him). But if there is a reason, such as if he 
    wants to go to another mosque to attend a lesson or for some essential 
    purpose, then there is nothing wrong with him leaving, even if the place for 
    doing wudoo’ is inside the mosque. End quote. 
He was also asked: I see 
    some brothers sitting in the mosque for a lesson, then when the adhaan 
    begins, they leave. I have read that the Messenger  (peace and blessings 
    of Allaah be upon him) forbade leaving the mosque when the adhaan is given. 
    What is the ruling on that? 
He replied: In al-Saheeh 
    it is narrated from Abu Hurayrah (may Allaah be pleased with him) that he 
    saw a man leave after the adhaan and he said: This man has disobeyed 
    Abu’l-Qaasim  (peace and blessings of Allaah be upon him). The scholars 
    said: It is haraam to leave the mosque after the adhaan without an excuse. 
    But this hadeeth does not clearly state that the man went out to pray in 
    another mosque. He may have gone out so as not to pray. What we think is 
    that if you go out to pray in another mosque where you are sure that you 
    will catch up with the prayer, there is nothing wrong with it, but that 
    should not be done lest others follow that example and go out without 
    praying. 
End quote from Liqa’ 
    al-baab il-Maftooh (9/38). 
Hence you will know that it 
    is permissible for you to go out of the mosque after the adhaan in order to 
    pray in another mosque where the imam recites better or to meet some of your 
    brothers there, if you know that when you go there you will be able to catch 
    up with the congregation. 
And Allaah knows best.