I have managed to complete the following 2 tasks:
- Get the R5C to boot from the sd card
- Copy the R5C DietPi installation from the sd card to the emmc
(1) How to get the NanoPi R5C to boot from the sd card.
In order to do this I executed the following command:
dd if=/dev/zero of=/dev/mmcblk1 bs=8M count=1
Or it was this (or very similar, I can’t quite recall):
dd if=/dev/zero of=/dev/mmcblk1p1 bs=8M count=1
This approach was obtained from Can't boot DietPi on the NanoPi R4SE - #2 by Jappe
Basically the objective is to break the default installed FriendlyElec OS, because the firmware will ignore the sdcard if it finds a working OS on the emmc.
Therefore, break the OS on the emmc so the firmware looks for a working OS on the sd card 
(2) How to get DietPi booting from the eMMC on the Nano Pi R5C.
After getting DietPi to boot I could run it on the sd card, but I wanted to run DietPi on the emmc. In order to achieve this I approximately followed this: Moving a running DietPi system to a USB stick/disk or an onboard eMMC – DietPi Blog
Noting that initially I spent a day trying to get that article work with an sd card that was larger than the emmc (the emmc is 32gb and the sd card i first tried was 64 gb). No matter what I tried I couldn’t get it to work.
So I just went out and purchased a 16GB sd card and then the following worked:
Step 1: Mark the / as readonly and reboot
mount -o remount,ro /
sed -i '\|[[:blank:]]/[[:blank:]]|s|,rw|,ro|' /etc/fstab
shutdown -r now
Step 2: Copy the sd card (mmcblk0) to the emmc (mmcblk1)
dietpi-services stop
swapoff -a
dd if=/dev/mmcblk0 of=/dev/mmcblk1 bs=1M status=progress
shutdown -h now
Then remove the sd card, and pull out the power and back in again to boot, and it should boot from the emmc.
Step 3: Return the / to read-write, and reboot and you are finished.
mount -o remount,rw /
sed -i '\|[[:blank:]]/[[:blank:]]|s|,ro|,rw|' /etc/fstab
shutdown -r now
(3) Further notes
During this work I found that Armbian works out of the box to install directly and automatically onto the sd card. It has a script build/packages/bsp/common/usr/sbin/armbian-install at 0a18ec690e423ffb25f66d39c09c0618f768e82d · armbian/build · GitHub
The script was useful to help me understand why Armbian worked and DietPi didn’t.