Sharegate supports migrating from Classic-mode authentication to a Claims-based authentication without any problem. However, it will not convert the authentication mode of your SharePoint environment. It will need to be set up correctly at the destination before the migration.
Sharegate will attempt to match users (or groups) from the source to the destination environment based on some properties that does not depend on the authentication provider. In other words, all these usernames:
- i:05.t|issuer name|user@domain
will be mapped automatically to the same user.
However, you can override this by manually mapping users and groups from the source to the destination to ensure that the user Sharegate uses is exactly the one you expect. You can get more information about the way Sharegate handles users and groups mappings here.
If you wish to change the authentication mode of your SharePoint environment from Classic to Claims-based (either SP2010, SP2013 or SP2016), you can find some information here.