No. Each process execution will have its own instance of the Document Cache. So, the Document Cache only lives for the length of a given execution and does not persist across the process executions.
However, the Process Call shape does not reset the cache. Calling multiple processes which use the same Document Cache component will load the cache with a mix of data from the different subprocesses. Always be aware of which Document Cache components are being reused when adding a Process Call shape.
For more information about the Document Cache component please see the User Guide: Document Cache components
Also check out this video that provides a deep dive into the Document Cache feature including fundamental concepts, component configuration, and practical usage: How to Use the Document Cache Shape.