Horizontal scaling with database schema

I am designing a database schema for our customers on-premise, which will be moved to Azure PaaS later. Regarding the "scale out" strategies I lean towards to create "one database", and create schema for each and every customer to store the database objects such as tables, procedures, and views. Although, I find little to no information how to keep track of schema changes in each and every schema, and put a version number on each. Tools such as Liquibase or Fluentmigrator has limited support for such it seems, as they seems to be better for a single database. What are the best practices to version, and manage database schemas in a database?