If you replicate a Linux VM right away you might end up with the serial console not working. This could be especially an issue if you have your network interfaces set as static and you have to change the IP addresses.
To enable the Serial Console you have to log into your on-premise Linux VM and run the following:
systemctl start serial-getty@ttyS0.service
systemctl enable serial-getty@ttyS0.service
Wait until the changes will be replicated, then connect directly from the Azure Console.