If a customer has a monthly membership that doesn't have payment information saved, or they would like you to change how they are billed on their behalf, you can add a new payment method to their membership.
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To do this go to their membership detail page (Customers >> Customer List >> Choose the desired customer >> Membership tab >> Choose active membership), click Payment method and you will see an option to add or change Payment Method.

You will then be taken to a page where you will be able to add new payment information and choose from the available payment methods.

These steps apply to child and family member memberships too. Each family member has their own unique customer profile and payment details need to be added to their membership individually, at this time.