Now, in the case of Cresent Tsukyomi's feat, I'll get that in a second. But first, I want to show another instance of Full Moon Tsukuyomi moving the moon. And it's in the very episode the OP linked for this thread.
(Timestamp, incase its not embedded, is 20:38)
What the OP happened to not show in his post here
Is that Misaki uses Full Moon Tsukuyomi and a moon appears behind her when getting ready to attack. Just like I showed above. And here, the moon is definitely shown to be moving. But don't take my word for it. The scans here, to ensure they aren't dodged or accidentally missed, will prove it for me.
The moon's positioning between these 2 moments, from the time Tsukuyomi attacks and by the time she finishes attacking, is blatantly not the same. We literally see the moon
behind Tsukuyomi when she's going into her attack phase, and then it's shown to be
in front of her, and her opponent, moments later when she's attacked. And we already know that Tsukuyomi actively calls the moon forth as part of her attacks, as the above clip shows is the case.
Why would the moon, during this specific attack phase, just randomly be in completely different positions if she wasn't moving it? The interpretation that it isn't, in this context, doesn't make sense and requires more evidence from the opposition to suggest an interpretation different from what the entire scene in and of itself already gives to us, and other supporting feats. More could be given for her, but I don't think I need to under these circumstances.