Is a specific mention (such as "tahlil" or "takbir") prescribed when accompanying a funeral to the grave, and what is required of those who follow and carry the funeral bier?

There is no prescribed act when following a funeral procession other than contemplating and reflecting on death and its consequences, and what comes after it, so that the follower may be prepared for what he will eventually face. What some people do, like saying "Remember Allah" or "Proclaim the Oneness of Allah," has no basis. It was the custom of the predecessors to remain silent and ponder the consequences of matters and the end of man, which necessitates taking a lesson. As for speaking, it is not beneficial in this situation.