And is a mention speeding the process up?

At end of day, for each slave in a guild there is a roll with a 20% chance to remove slaves from the top, so the more slaves that are in a guild the faster they are removed. Though in general it's best to pick one slave guild to sell all your slaves and buy from the the others.

If a guild has less than 4 slaves, then it gains one slave. If it has less than 6 slaves, there is a 75% chance to gain another slave.

Ending the day more, will result in more changes. Otherwise, buy all but 1 of the slaves in a guild(Umbra is all but 3), then save and reload, it will be restocked with 4-6 slaves.

Selling slaves to the guild will put them into list of slaves you can buy there. That can clog up the list. If you go into the control window in Town you can quicksell captives for much less money but they will not reappear in the slave guild.