As community members, you can change their email address on the profile page of the community. In the 'Email' field they can change their e-mail address and click 'Save' click. After this, they will receive a confirmation message at their new email address. There they must confirm their e-mail address. If they don't then stay they are known under their old e-mail address and use it to log in. A community member therefore no longer needs to have access to his/her old mailbox, the confirmation message goes to the new email address.
In order for the e-mails for changing an e-mail address to arrive properly, a sender name and sender address are set. You can do this in Community in the 'Email' menu. Do you use your own domain to send the emails? Make sure the SPF record set state. If you do not use this, there is a greater chance that emails in the spam ends up.