I know VainMain's answer from above is probably more careful and thorough, but I've been able to successfully move the partion by simply doing:
In Windows 10: "reagentc /disable"
In Linux boot CD: adjust neighboring partition as needed/move recovery partition
In Windows 10: "reagentc /enable"
Recovery environment was automatically rediscovered and booted WinRE just fine with all recovery options (Reset/System Image restore/etc). I had only 1 C: partition, no special partitioning/dual booting/multiple recovery partition/crazy BCDEDIT settings beforehand, which helped. Tested inside a VM beforehard to make sure. Had no problems after executing live.
If I remember correctly, skipping the first step (disabling via reangetc) would cause the recovery environment to end up misconfigured, not properly re-bootable, and not easily fixed.
