Thanks for your efforts, Daniel…
One thing I noticed - on the Github issues, you marked this topic as “Opi Plus”. I have an OPi PC Plus - so i’ts not exactly the same board (although DietPi installs and runs quite nice, thank you… 
Also, I’ve successfully installed Armbian 5.25 (server version, no desktop) to emmc, using the armbian “nand-sata-install” script: the script offered me the option to “boot from emmc, run from emmc” (or something like that), and I chose that and started the emmc copy.
After a while, the script installed the system to emmc, I shut the system down, took out the SD card, and booted from emmc without any issues.
BTW, kernel for that Armbian server (5.25) I have is:
root@orangepipcplus:~# uname -a
Linux orangepipcplus 3.4.113-sun8i #10 SMP PREEMPT Thu Feb 23 19:55:00 CET 2017 armv7l GNU/Linux
So, I somehow suspect that it’s not the kernel issue… since this arbian version uses the same kernel.
As for your final suggestion, that is exactly what I did: I used armbian SD image to boot the Opi PC Plus, and copied the DietPi image (which I had copied on an external USB flash) to the emmc. The copy went without any issues, but when I took out the Armbian SD card, and tried to boot from emmc, no dice… - although the complete (working) DietPi image was copied to emmc (using “dd”) 
I also marked the boot partition as bootable, just in case (using gparted in armbian desktop booted from SD card), but that didn’t help, either…