1. A system, comprising: a plurality of compute nodes that together implement a provider network that provides virtual computing resources to clients and a network-based interface configured to process requests for the provider network, wherein individual ones of the plurality of computing nodes are located in different ones of a plurality of availability zones implemented as part of the provider network; one or more of the plurality of compute nodes that together implement a migration manager for the provider network, configured to: receive a request to migrate a virtual compute instance hosted at a compute node located in a particular availability zone to another availability zone of a provider network, wherein the virtual compute instance is currently operating for a client of the provider network, wherein the request to migrate the virtual compute instance is received from the client via the network-based interface, and wherein the particular availability zone and the other availability zone provide geographically diverse, fault-tolerant zones for hosting respective virtual computing resources such that in the event of a failure of one of the particular availability zone or the other availability zone the respective virtual computing resources hosted at a remaining availability zone continue operation; configure a destination virtual compute instance hosted at another compute node located in the other availability zone that is provisioned for migration of the virtual compute instance, wherein the configuration is based, at least in part, on a current configuration of the virtual compute instance; complete the migration of the virtual compute instance such that the destination virtual compute instance is currently operating for the client of the provider network, wherein the virtual compute instance is not currently operating for the client of the provider network; and send a notification to the client indicating the completion of the migration for the virtual compute instance.