Thank  you for including the images. The moss on the branches is likely growing there because the healthy, dense top growth is providing the shade and moisture moss needs to thrive. Moss commonly develops on slow growing plants but doesn’t damage the plant it is growing on.

You can kill the moss on the branches but it is likely to return because of the favorable conditions – shade and moisture. If you decide to treat, choose the correct product, one that can be used on plants (Products for hardscape, such as walkways and decks, will seriously damage your crepe myrtle.)

Formulations are either liquid or granular; liquids provide the best coverage. The treated moss will turn black and can be brushed off or be allowed to weather in its own time frame.