I have 2 rancher servers, and I want to move downstream clusters from one to another. What would the steps be to disassociate a downstream cluster with server A and register it with server B, while retaining the deployed applications in the cluster?

Was the downstream cluster created from Rancher UI or imported into Rancher ?

Created with the API, and then docker commands were run on instances to join and register with the cluster

There is no way to transfer between Rancher Servers. You can "import" the old cluster into the new, but you will lose a lot of management abilities. I've pretty much always gone the route of simply building a new cluster and transferring the workloads.