In order to update the percent complete in your native programme file you have to Create a custom report at the project or company level that includes specific task details (i.e. Activity ID, ID, Name, Is Summary Task, Last Updated and Percent Complete). See Create a Custom Project Report and Create a Custom Company Report.
Once you create a report, apply a date range filter depending on how frequently you update your native or master programme. Any 'last updated' programme tasks within that date range will include programme tasks that had the '% complete' field updated. You can distribute that report on the cadence you update your programme and use the data from the report to determine what changes need to be made to your native programme file.
Note: Updating the percent complete in the Procore mobile app does not update the percent complete in the native programme file.