I'm not really sure, but the following paragraph in the rules suggests otherwise:
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.
If they were assigned at the same time (both FP player and shadow player), then the situation above wouldn't be even possible, as no effect would be playable between FP player assigning and shadow player assigning.
To me that only precludes the possibility that a minion that becomes "unassigned" during the assignment phase from becoming a leftover minion and able to be assigned by the shadow player.
I can see where you are coming from, however since it does specifically mention timing, that "all assignments take effect simultaneously" I would argue that there is only one trigger for
DDotR at the end of the assignment phase. Otherwise it should trigger with each minion that is assigned by either the shadow player or the free people's player.
Here is how I believe that Decipher intended it:
Assignment actions take place, that can only result in one-on-one assignments.
Once both players pass the free people's player assigns any currently un-assigned minions (that they want to) to remaining un-assigned companions (accounting for Defender bonuses)
Any leftover minions may be assigned by the shadow player (if they want)
Once the shadow player indicates he is done, the assignments all "latch" at the same time and any "each time" assignment triggers happen then.