I don't think Azure has a live migration. In fact, out of Google Cloud Platform, AWS and Azure only GCP has live migration of the running VMs.
If users will favour GCE because of reduced downtime support due to the live migration support (or simply favour GCE because of they perceive it as a more advanced platform), then other providers will follow.
Right now it seems that it's all a price war, not yet a feature war. But prices can go down only to a point, then feature war will start showing it's effects, and it will be interesting.