In this sort of fairly linear situation, where in each event along the way the heroes implied goal is "get through this as fast as possible" and the relevant opposing "goal" is "slow them down until everything caves in", I think you could easily adapt the heroquest mechanic.
There are a number of contests that they face, and in each one there will be carryover depending on their results. So an early loss means something like one of them gets pinned by collapsing rubble and the others have to pull him/her out (taking up time). Similarly there could be contests to not get lost or slowed down in a maze like section of tunnels, a contest to squeeze through a tight spot, and of course, one or more fighting the Karshtids. One cool contest could be they see see the little chaos-bugs chewing through a natural rock pillar that seems to be supporting the roof--do they stop to kill them, use magic to reinforce the rock, just run past as quickly as they can?
You can either use the incremental carryover described in HW (each step affects all following steps), or else save the carryover to affect only specific other contests, like the final one where they make the sprint towards daylight as everything begins to give way....
For what it is worth, I think the carry over mechanic has interesting uses far beyond hero quests. This is already recognized informally when contests are written up with notes like "If they were offered meat at the greeting, they get +5 here. If they failed to even get offerd a blanket they will be at -10 here." Using the carryover mechanic just formalizes this a bit.
--Bryan
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