I don't think the Meka Dragon example power up has "on your turn" in the text, does it?
"Discard this Evolution on your turn if you are not in Tokyo. Skip your turn but..."
The Kraken version (Sunken Temple) is a Permanent version, but doesn't have the 'on your turn' bit:
"If you are not in Tokyo you can skip your turn to..."
Either way, it pretty much has to be your turn in order for you to use the cards in order to skip your turn.
So you get a turn, but you don't get a turn on your turn.