Hello!

Do these monsters stay in the same place if they are with an investigator on the monster movement part of the mythos phase?

No.

Tibs said:

So they move and do not stay with the investigator? Where is it written? Many thanks!

This is from the author:

KH] 1. Do Elusive Monsters still not move if an Investigator is in their space? Given that they deliberately avoid combat, it seems that Elusive monsters should move whether an Investigator is in their space or not. Otherwise it's a pretty big weakness, since a player could neutralize Tulszhca's Elusive Cultists just by staying in their space. And do Hastur's Cultists NOT swoop down on Investigators, but only on Elder Signs when Tulzscha is the Herald? All of this would seem very logical, but the rulebook doesn't say. (06/13/08)
Ok, I've got some time this afternoon, so let me try to clear these up.
Elusive monsters will ignore investigators for purposes of movement, so they'll move away if they're in the same area as an investigator. This will be errata'ed when we update the FAQ for Kingsport.

http://www.fantasyflightgames.com/edge_foros_discusion.asp?efid=5&efcid=1&efidt=895

Ah, so specifically, they ignore investigators for movement purposes? That would mean that Glaaki and Tulzscha's elusive stalking Cultists are effectively elusive black-bordered monsters. Good to know! I don't know why I didn't piece that together sooner. Thanks, mageith .

thanks to you all!