Do future start dates of subscriptions migrated from an external application get activated automatically?

Future dates of migrated subscriptions don't get updated automatically. However, when you activate a subscription before the start date, the dependent transactions have the respective start dates for reference and aren't impacted by the migration. And when the actual dates are reached, related revenue and Accounts Receivables invoices are processed.

Or, you can use the Pending Activation feature to keep the subscription from actual activation, and based on the start date you can activate it by removing the Pending Activation mark.