What is prescribed for the one who is praying behind an Imam is to follow his Imam, not to do anything before him or at the same time as him, or to delay doing it. Rather he should do things straight after his Imam does them.
This is indicated by the report narrated by al-Bukhari and Muslim from the hadith of Anas ibn Malik and Abu Hurayrah (may Allah be pleased with them), according to which the Messenger of Allah (peace and blessings be upon him) said: “The Imam is only appointed to be followed, so when he says takbeer, then say takbeer; when he bows, then bow; when he says Sami’a Allaahu liman hamidah, then say Allaahumma rabbana laka’l-hamd. If he prays standing, then pray standing.” (Sunan Abu Dawud).
Thus, delaying in following the Imam contravenes this prophetic injunction. It may even lead to the invalidity of the Prayer if the follower (ma’mum) becomes two rukns (prayer pillars) behind the Imam for no excuse.