This only applies to Linux images in Docker for Windows.

I was able to move the MobyLinuxVM through Hyper-V manager and update %APPDATA%\Docker\settings.json with

"MobyVhdPathOverride": "E:\\Hyper-V\\Virtual Hard Disks\\MobyLinuxVM.vhdx"

Even after multiple attempts, Docker would re-start in the old directory and create a new virtual hard disk and over-write settings.json.

It turns out you also have to re-configure Hyper-V to point to the new folders:

Once I updated Hyper-V and re-started Docker, all was as expected.