well before you get ahead of yourself, think about your initial description of the problem "Lift runs for a while then when it gets hot it just dies"

is it running hot?
make sure its not a heat related problem with overheating first.
an overheating truck will shut down if it hits stage 2 overheat.
(generally you will have overheat codes though)
have you checked the fault history to see if there are any?

otherwise i'd go with what kevin suggested.
a faulty connection or ground or maybe it could be a relay or ecm related problem showing when it warms up.