In my last game, I had an activated CSA Kentucky militia...and I moved it to Lexington. And I held my breath and...nothing happened!
So aha, I think...it's ok to move Kentucky troops in Kentucky, and not have it be an invasion (at least I think)...hmm, good to know.
A turn later, I decide to be bold again, and move the same militia to Union held Bowling Green. Again, no invasion, yay!
The turn after that, I move a CSA Kentucky conscript within Kentucky. It USED to be a militia, but it trained up. BOOM...Aaaargh! "CSA Invades Kentucky!!!"
Sigh.

Rats. OK, now there are two explanations for this, I think: 1) moving a Ky non-militia within Ky will trigger invasion (but "pure" militia ok?); and/or 2) the militia I moved to Bowling Green began siege operations against the Union Ky militia stationed there...might that trigger invasion?
Kentucky activation...please help, I MUST unlock it's mysteries!
