Thanks for the reply.
I was deactivating all the automatic updates to avoid different versions in the migration. After posting here my question, I’ve contacted SiteGround support that promptly assisted.
Even taken longer than I expect, migration was concluded with success. The negative impact was a downtime longer than expected.
Suggestion: Include in the handbook for SG Migrator certain operational tips that may help a smoother and planned process, namely:
-memory adjustment: at the end I’ve got an error and SiteGround support indicated to adjust memory in previous host… and you have a file already with instructions how to do it.
-advising to upgrade first WordPress and Plugins (allowing to make in a more planned and controlled way before migrating).
-advising to deactivate plugins for a faster process.
These tips would had cut drastically my migration process and the downtime, as it would work at first attempt and not at third with a reverse to previous WP version in old host…
In all the process SiteGround support team answered immediately. Congratulations for that.