What is the job title of the man who slowly walks in front of a funeral prossession?

I believe in the UK that it is the funeral director who will walk in front of the procession.

Every funeral is different. Sometime a mourner will lead the procession.
A mounted escort led the funeral procession in Hobart, Australia for Alec Campbell,
the last veteran of the Gallipoli campaign.
However, in most parts of North America a hearse leads it. This tradition and custom
can be traced back to African roots of the Bakongo and the LaDogaa tribes.
http://www.pbs.org/pov/homegoings/african-american-funeral-director/

quatarexpat is correct in respect of UK funerals.

https://www.quora.com/Why-do-funeral-directors-walk-in-front-of-the-hearse

Most of us will have an enduring and solemn image of this duty being carried out by a suitably funereal looking Rodney in a classic episode of 'Only Fools and Horses'.

In Dickens' Oliver Twist, the boy Oliver does a gig leading funeral possession - but he doesn't like it much and soon stops doing it.

I always heard that it was called the Mute and people were hired to do this and yes, this was Oliver Twist's job with the Sowerberrys before he hopped it.

These sites call them the funeral conductor. They are carried out by the funeral director as described above, but in the capacity of a funeral conductor or officiant.

https://www.everplans.com/articles/what-does-a-funeral-conductor-or-funeral-officiant-do