All perfect praise be to Allah, The Lord of The Worlds, and may His blessings and peace be upon our Prophet Mohammad and upon all his family and companions.
Muslim scholars have agreed that sending prayers upon the Prophet (PBUH) after the Adhaan is desirable because he (PBUH) said: "If you hear the Muadh-dhin then say as he says. Then send Salat upon me, because whoever sends Salat upon me, Allah Wll Send Salat upon him ten times due to it. Then ask Allah that He Gives me Al-Wasilah, because it is a place in Paradise which is not for anyone except for a slave from the slaves of Allah, and I hope that I am him. And whoever asks that I have Al-Wasilah, then (my) intercession will be made lawful for him." {Muslim}. This Hadith includes the Muadhin and those who hear the Adhaan. "It is permissible for both the Muadhin and those who hear the Adhaan to send prayers upon the Prophet (PBUH) after the Adhaan ends." {Moghni al-Mohtajj vol.1/pp.219}. "It is desirable that the Muadhin, after the Adhaan ends, sends prayers upon the Prophet and asks Allah to give him Al-Wasilah." {Al-Majmou` vol.3/pp.117}. " Then, after he finishes the Adhaan, he sends prayers upon the Prophet (PBUH) and says: "Allahumma Rabba hadhihi-dda` watit-tammah,' [O Allah! Lord of this perfect call (perfect by not ascribing partners to You)……" {Shareh Montaha Al-Eiradat vol.1/pp.130}, a Hanbali book. However, the Muadhin`s sending prayers loudly through the external loudspeakers after he finishes the Adhan is usually highlighted in this regard, whether the volume was loud or not. This used to be done by many a Muadhin (person who calls for prayer) over the late centuries throughout the Muslim territories. The majority of the late Muslim scholars considered sending prayers upon the Prophet "loudly" as permissible or desirable for they believed that it doesn`t involve any violation to Islamic Sharia and it doesn`t contradict the purpose behind the Adhaan since it reminds people of this Sunnah (sending prayers upon the Prophet) and spreads it in the open. Therefore, {Hasheyat al-Dasookhi Ala Al-Shareh Al-Kabeer vol.2/pp.220}, a Maliki book, stated: "As for sending prayers upon the Prophet (PBUH), it is a good innovation." Moreover, the Hanafi scholar Ibn Abdeen stated: "The correct opinion of the scholars is that sending prayers upon the Prophet is a good innovation." {Radd al-Mohtaar}.
Al-Imam al-Sakhawi stated: "…………….scholars have disagreed whether sending prayers upon the Prophet is desirable, reprehensible, innovative or lawful. The first opinion was based on the verse: "and do good." [Hajj/77}. It is axiomatic that sending peace and prayers upon the Prophet (PBUH) is an act of devotion that brings a Muslim closer to His Lord; particularly, since the Prophetic tradition urges Muslims to do that. This is in addition to the blessing of sending prayers upon the Prophet (PBUH) and saying the Dua`" 'Allahumma Rabba hadhihi-dda` watit-tammah,' [O Allah! Lord of this perfect call (perfect by not ascribing partners to You " after the Adhaan, at the last third of the night and prior Fajr (dawn) time. The correct opinion is that it is a good innovation for which a person is rewarded based on his intention behind saying it (e.g getting closer to Allah)." {Al-qawol Al-Badei` Fi Asalat Al Al-Habeeb Ash-Shafei`}.
However, it is better that the Muadhin pauses between the end of Adhaan and sending prayers upon the Prophet in addition to the above Dua` in order to show that it isn`t part of the Adhaan, and this is the practice observed nowadays. In conclusion, it isn`t permissible to turn this side issue into a cause for conflict and dispute at mosques since, in principle, people should follow the practice observed by the Ministry of Awqaf, and make no objection about matters of dispute. And Allah Knows Best.