Does the Imam say the takbir before moving or at the end of the movement, and what is the ruling on congregants preceding the Imam in movements because they start them upon hearing the takbir and not seeing him?

It is recommended for the worshipper to begin the Takbir (pronouncement of "Allahu Akbar") at the start of entering a new posture, such as bowing (Ruku') and prostration (Sujud). So, if the worshipper raises his head from prostration, he should pronounce the Takbir at the beginning of raising it and finish when he stands upright. If the worshipper sits for the resting-sitting (Jalsat al-Istiraha), the Takbir ends when he finishes his sitting, then he rises to stand without Takbir. As for the follower preceding the Imam in some transitional Takbirs, it does not invalidate the prayer according to the majority of scholars. However, preceding the Imam in the opening Takbir (Takbirat al-Ihram) invalidates the follower's prayer.