I mean, it could be cool, but I don't really grok the idea of Lunars touching things and transforming their bodies.
I think it makes perfect sense, assuming that 1) Lunar Charms are godbody Charms, and 2) that the body of a god is more than base flesh.
So, a Lunar might transform her mount or familiar not because she has the power to transform things, but because she can transform herself, and her most beloved companion is as much part of her as her arms or heart.
Or, conversely, she might have an iconic steed that she always rides, even when she rides something else (briefly turning any random mule into Sleipnir or Kintou'n
Think of the vÄhana
of Hindu deities, which are less separate entities and more attributes and symbols of the riders; Luna riding the Silver-Horned Watcher, if you will.