The data migration will result in new data being entered into your destination app - which means, if you have any triggers or automations (including notifications, workflows, and so on) enabled, these may be set off.
While some tools have special workarounds in their API capabilities to bypass triggers (for example, Zendesk), many do not. It will be up to you to ensure that any triggered actions you do not wish to set off are disabled in your destination app before the migration begins. This can include:
user notifications that data has been assigned to them
workflows that update or create new records, send emails, etc.
customer notifications that their data has been updated (especially in the case of support tickets)
and more
If you are unsure, we recommend reaching out to your app support or reviewing their support resources to understand what may be triggered when data is added to their app.