The giant altar was moving it at first, but when the altar got destroyed, the moon stopped moving. However, when Toneri fully awakened his own tenseigan, he was able to restart the moon's crash course, so yes the tenseigan can move the moon.
OT: The reason why the evolution(s) of the byakugan aren't as covered as much by fans despite their power, is because it takes very specific and rare conditions (that normal Hyuugas in general could never hope to achieve) in order to awaken these eyes.
Awakening tenseigan is essentially like awakening the rinnegan, and we know that most Uchihas can't even awaken the rinnegan. Even the two Uchihas who did awaken it only got the chance to awaken it under very special circumstances (and both had just nearly died when they awakened it, and both are transmigrants). The tenseigan is basically the same way with the Hyuuga/Otsutsuki.
As for the jogan, we don't even fully know what that is yet. Boruto might have just gotten it due to the parents he was born to (and luck) rather than it being an evolution that a normal Hyuuga could achieve.