Yes. You don't see it happen as often as with Rainbow Energy, of which I'm sure, but it works the same.
The text of Beast Energy:
"This card provides 1 Colorless Energy.
While this card is attached to an Ultra Beast, it provides every type of Energy but provides only 1 Energy at a time. The attacks of the Ultra Beast this card is attached to do 30 more damage to your opponent's Active Pokémon (before applying Weakness and Resistance)."
Text of Rainbow Energy:
"This card provides Colorless Energy. While in play, this card provides every type of Energy but provides only 1 Energy at a time. When you attach this card from your hand to 1 of your Pokémon, put 1 damage counter on that Pokémon."
Since both could be a lightning (every type but only 1 at a time), they could both move to Tapu Koko GX. The Beast Energy becomes indeed a colorless energy, the Rainbow Energy will still be rainbow, so every type but only 1 at a time.
Fun fact, if you would use a second Tapu Koko GX and moved the Beast Energy to the first Tapu Koko GX, then only the Rainbow Energy will move with that ability since the Beast Energy has become colorless (not attached to an Ultra Beast anymore).