I exported the .vcf from Moto G to Google Drive.
Importing a .vcf wasn't available yet for the new Google Contacts (contacts.google.com), so they showed steps to go to the old Google Contacts.
After importing, I merged to get rid of the duplicates.
After doing this, changes on Moto G were reflected on the desktop https://contacts.google.com/preview/all, and other mobile devices.
I don't think I changed any sync settings on the Moto G to get it to work, but it's possible that something else solved the problem, instead of importing from Moto G to the desktop Google Contacts.