"I have x identical database schemas, one per client "
and
"I also have models for each one of these databases"
Did you mean I also have models for each one of these Schemas?
Ef model entities are linked to a Schema/TableName.
entity.ToTable("tableName", "schemaName");
So if each client gets their own schema, then each model has all tables in that schema and the client gets their own model. So how do I run update-database on each Context model. I another way of looking at the issue.
So the answer lies with how you are tracking the schema per client info.
Powershell migrate.exe approach might be interest for your so that you can trigger migration on many context models.
Custom migration operations might also be interesting. Rowan is an EF developer.
EDIT: based on Auto Migration comment, this is worth a look Managing migration triggers in code