This is an interesting thread. My '20 XLT FX4 MAX has 3.5 years and 45K miles on it. During that time I have had these messages appear 3 times. The first two times, I had been in one of the 4x4 drive modes for an extended period of time/miles due to inclement weather/road conditions. At some point I assumed that the various modes set some sort of default ELSD "clutch pressure", rather than just briefly turning it on and off for yaw control or wheel slip, etc. and that long term pressure was (over) heating the ELSD hydraulic unit. After coming to this conclusion I started simply pushing the A4WD button in inclement weather while keeping the truck in "normal" drive mode. I've done that many times and never experienced the the dash alerts. This has led me to believe that while various drive modes activate the ELSD at some constant, default level, putting the transfer case into A4WD while still in "normal" drive mode does not affect the ELSD "normal" operation. Once again - I'm guessing.
The third time that I experienced the warnings was this fall during an offroad experience described on another thread disparaging the flocked fender-wells on these vehicles. At the time I had the truck in "mud and ruts" mode with the differential "lock" button engaged for a very short amount of time, say 1/4 mile at crawling speed. Note that I can only remember hitting the diff lock button a couple of times during the life of the vehicle. Due to mud build up described in the other thread, the truck was shut down for more than an hour immediately after the event. When the truck was restarted, the wrench and all warnings were gone. I was surprised at just how quickly everything heated up. The ambient temperature was close to 32 deg F at the time.
At some point, I plan to check the hydraulic (ATF) level in the ELSD hydraulic unit. Might be that the level is low, contributing to the rapid heat up.
I don't believe that any of my issues relate to low battery - not that mine doesn't experience low battery levels, it does - If I have several weeks of short duration trips, especially in the winter months when headlights, seat heaters, etc. are all on, it will take the battery down and different items will shutdown; but, when I've experienced these particular drivetrain warnings the vehicle has experienced long periods of operation and therefore had high battery levels.
That being said, another contributing factor to hydraulic system heat could be higher current draw under low voltage conditions such as others have described.
Bottom line, these hydraulic units seem to overheat under conditions that, as drivers/customers, we consider to be within the realm of normal operation - which is a problem by definition.