If I want someone to stop receiving emails from one flow and instead receive emails from another, how can I do that?
The best way to accomplish this would be to add an "Update Profile Properties" element to both of the flows, before the first email. After dragging in the Update Profile Property action, select Create New Property from the first dropdown. Give your custom property a name (e.g., Flow), and enter a value (e.g., Welcome Series, Birthday, etc.).
Next, add an Additional Filter to all of the emails in both Flows. To do that, on the email, select the funnel icon on each email where you want this condition applied. These Additional Filters will check the value of the custom property set above. If the value matches the name of the flow, then the email will send, otherwise, it will skip. In this instance, the configuration of the additional filter would check Properties about someone where Flow equals Welcome Series or Birthday.
This way, if a Profile is added to the Welcome flow, the Flow custom property is set to Welcome Series, and each email within that flow is configured to check that the Flow custom property equals Welcome Series. Now, a little later, if that same Profile is added to Birthday, the Update Profile Property action in the Birthday flow overwrites the Flow custom property from Welcome Series to Birthday and now the remaining emails in the Welcome Series will fail the Additional Filter check. However, the Emails in the Birthday flow will send, since the Flow custom property matches the Additional Filter in that Flow.