From the comp rules:
"The creation of leftover minions happens only when the Free Peoples player is done making assignments. If an assigned minion subsequently becomes unassigned as a result of a card's game text (e.g. a triggered ability), it is not a leftover minion and is not eligible to be assigned by the Shadow player."
Tricky word here is "subsequently". Does it refer to the time after the FP has done making assignments or does it refer to the time after the minion in question is assigned, but not before the FP ended assigning.
If it's the former, then indeed, due to the above mentioned rule the minion in question is not a "leftover" minion. However, if it's the latter, then the FP player still can assign that minion and it can be "leftover" only when the FP concludes his part of assignment.
I believe it's the latter, so in case of
Desert Runner killing a companion via his text during the Assignment Phase, then
Desert Runner is unassigned and the FP has the choice to assign him or leave him for the Shadow Player to assign.