Move Azure website between App Service Plans
The original accepted answer is more than a year old. In Azure world, that's an eternity. The "Change App Plan" button in the Azure Portal is no longer present it appears (as of 2016-06-17), so I'm throwing in the Powershell command I needed to use in order to move a Webapp to another App Service Plan.
Set-AzureRmWebApp -Name <webapp name> -ResourceGroupName <resource group name> -AppServicePlan <new app service plan>Set-AzureRmWebAppSlot -Name <webapp name> -Slot <slot name> -ResourceGroupName <resource group name> -AppServicePlan <new app service plan>
Yes, you can do this from the Web app blade in the Azure portal. You have to expand the toolbar though to see the option. See below.