Is it a condition for the Imam to say the Takbir when moving from the first Tashahhud to the third Rak'ah for the congregants to follow him, or is it sufficient for him to merely begin to move?

It is recommended for the worshiper to begin the transitional takbeer (Allahu Akbar) at the start of performing the rukn (pillar or major part of prayer). The follower should emulate the Imam in both the takbeer and the action simultaneously, starting after the Imam begins and before he finishes the rukn. If the Imam is visible, then what matters for emulation is his action, not his voice. As Al-Bara' bin Azib said: "We used to pray behind the Prophet, peace and blessings be upon him, and when he said, 'Allah hears those who praise Him,' none of us would bend our backs until the Prophet, peace and blessings be upon him, placed his forehead on the ground." Sheikh Muhammad bin Salih Al-Uthaymeen mentioned that what matters is the Imam reaching the ground in prostration, not his voice. So, if you know that the Imam reaches the ground before his voice ceases with the takbeer, then prostrate.