Is it true that whoever responds to the Mu'adhdhin receives the reward of every man who prays in the mosque, and is this reward linked to the mosque in which the respondent prays?

Many hadiths have been narrated encouraging the answering of the muezzin and clarifying the virtue of those who do so. Among them: On the authority of Abu Sa'eed Al-Khudri (may Allah be pleased with him), who said: The Messenger of Allah (peace be upon him) said: "When you hear the muezzin, then say what the muezzin says." On the authority of Abdullah bin Amr bin Al-Aas (may Allah be pleased with both of them), he heard the Prophet (peace be upon him) say: "When you hear the muezzin, then say what he says, then invoke blessings upon me, for indeed whoever invokes one blessing upon me, Allah will invoke ten blessings upon him. Then ask Allah for me Al-Waseelah, for it is a station in Paradise that is not fitting except for one servant of Allah, and I hope that I am he. So whoever asks Allah for me Al-Waseelah, intercession will be granted to him." On the authority of Umar ibn Al-Khattab (may Allah be pleased with him), who said: The Messenger of Allah (peace be upon him) said: "When the muezzin says 'Allahu Akbar, Allahu Akbar,' and one of you says 'Allahu Akbar, Allahu Akbar'... [until the end of the adhan with the mentioned changes]... then he says 'La ilaha illallah,' and says 'La ilaha illallah' from his heart, he will enter Paradise." On the authority of Abu Hurairah (may Allah be pleased with him), who said: "We were with the Messenger of Allah (peace be upon him), and Bilal stood up and called the adhan. When he finished, the Messenger of Allah (peace be upon him) said: 'Whoever says what this one said with certainty will enter Paradise.'" On the authority of Abdullah bin Amr (may Allah be pleased with both of them), a man said: "O Messenger of Allah, the muezzins have a preference over us." So the Messenger of Allah (peace be upon him) said: "Say as they say, and when you finish, ask, and you will be given."

There is no indication that whoever follows the muezzin will have the reward of one who prayed with him. Rather, a hadith states that whoever follows the muezzin will have a reward similar to his, and this is a reward obtained by everyone who answers the muezzin, whether he prays with him or prays in another mosque.