Questions about migrating VMs (with virt-manager)

Hello, I want to migrate multiple VMs to a new host. In the process, several questions came up, where I could not find an answer.  I am using virt-manager version 1:5.0.0-5+deb13u1. I reordered and renamed my storage pools: When migrating a VM, how can I move the VM to a specific storage pool? Will the VM be moved to a storage pool with the same name by default? When migration finishes, the VM on the source host will still be accessible but will shut down. Is that true? Can I prevent the VM from shutting down during the migration? Thank you.