I know nothing about kombucha. (Doesn't it use a big SCOBY mother gelatinous blob?)
But when re-using beer yeast, I try to do one of the following:
Good - Save yeast after closed transfer in the fermenter. Try to use within a week or two.
Better - Save yeast in a sanitized mason jar at refrigerator temperatures. Can survive for months/years depending on generation # and sanitary practices. Needs a starter if not used within a week or two.
Best - Re-pitch immediately.
Were I a kombucha drinker, I'd probably wait to transfer off the mother until I'm ready to pitch the next batch. (Is it tea and sugar? Really stretching the ol' memory here.)