Great question;
are you referring to VSET ability "Mount", or like Transport from Troop Cart?
I think mount took a 'replaces-attacks" to mount or dismount.
I am assuming,
you had a character 'onboard' a mounted/ transport figure, that was defeated over a pit?
Hothie and SithBorg and Jim-2 etc
and some folks from the Wizards of the Coast forum era,
had different interpretations over how to handle the reconciling of order of operations on that.
FlyingArrow and Ugor and a few other folks had some great points.
Steve-O made a few counter-scenarios, it's somewhere in the Wayback Machine IIRC...
I think a lot of the Troop Cart discussions would apply here.
If So,
I think the only relevant scenario is,
when dismounting over a pit.
If you do not have flight, I don't think you can dismount over a pit as that would be insta-defeat.
Local leagues allowed the "Roll D20, if < 11, take 20DMG and moved to adjacent to Pit" as a fair house rule in
scenarios where figures on a transport that is defeated over a pit are 'ejected' from that when the model is defeated.
So, in that interpretation - they do not check for Special Ability among the Container of the Carried/Transported Inventory Management List
The Rulebook term "Character" is used instead,
to allow a 'Avoid Defeat" style roll for any character that would otherwise be defeated when
I think it's fair for a "Avoid Defeat" style rule for embarked characters to be moved to a adjacent-to-pit-edge square.
Just a mod to the Startup Placement Rules for placing characters on the battlefield really.
Its like Space Hulk anyways so, I don't see the issue for Home Games for that House Rule / Venue Specific Floor Rule to allow that.
The other interpretation is
"Embarked Characters are like Reserves 20 or Reinforcements etc, so, they are not actually still "In-Play"
they are moved to a phase at the end-of-turn consolidation in the OrderOfOperations.
Therefore, they do not have an activation to expend to leave being 'transported'
and so, they are stuck 'wherever Out of Play" is.
The victory points are inclusive of any player transported aboard.
yes, I have seen arguments where people split hairs over "Is the Game Concluded if , I had transported' characters on a troop cart etc
that are defeated over a pit, are all pieces defeated for the purposes of
Concluding Victory Point?"
That was a 'fun' arbitrage by the Chess Time Principle in the
Floor Rules Round-Robin Terms and Conditions etc...
Technically, if there are pieces in a "Out of Play" area,
are they all defeated? No.
But, most games don't go by "defeat all opponent pieces" as the main or sole criterion.
It is a check of the Victory Points at the Time. A regulation Game is what, 60mins max?
many locales have 30mins AND use a Chess Clock for the time so that
Slow Play is TKO Prevented