No, you should not assume that your old LSA leads will move into the migrated Google Ads campaign as new leads. Migration generally moves the campaign and its settings, while historical lead records and reporting may remain tied to the original LSA account or reporting system.
Before the migration, save your existing lead data, including lead dates, contact details, lead types, disputes, and outcomes. This gives you a reliable record to compare against the new campaign.
After migration, also check how new leads are being recorded and whether your CRM or call-tracking setup is still working correctly. That way, you can keep your historical data separate while accurately tracking new leads after the transition.