Azure: Moving an App Service to another existing App Service Plan
Not sure if it is something recent, but it seems it is finally possible. The option is in the main resource menu as "Change App Service plan". See:
I was able to move an app service(website) and also its deployment slot to a different plan (with same parameters and location - but did not try if this is a requirement).
Yes, we can now move the App Service Plan using the Change App Service Plan.
But sometime you will get No App Service Plan Found as this is the known restriction for the moving App Service plan.
Restrictions
- We can move the App service plan only if the new plan is in the same resource group and geographical location.- These two conditions is is to ensure the new plan is in same webspace
Azure deploys each new App Service plan into a deployment unit, internally called a webspace. Each region can have many webspaces, but your app can only move between plans that are created in the same webspace. An App Service Environment is an isolated webspace, so apps can be moved between plans in the same App Service Environment, but not between plans in different App Service Environments.
You can’t specify the webspace you want when creating a plan, but it’s possible to ensure that a plan is created in the same webspace as an existing plan. In brief, all plans created with the same resource group and region combination are deployed into the same webspace. For example, if you created a plan in resource group A and region B, then any plan you subsequently create in resource group A and region B is deployed into the same webspace. Note that plans can’t move webspaces after they’re created, so you can’t move a plan into “the same webspace” as another plan by moving it to another resource group.
You can find this in the official doc here and here
You can also refer this SO
The reason why you are not able to see the other App Service Plan when you try to change the App Service Plan is due to current limitations in moving the Azure web app resources.
The other App Service Plan which you intend to move your Web Apps to is in another resource group with existing Azure Web Apps, which is not supported currently
Azure Web Apps current move limitations updated 01/04/2016
Hope this answer your question.