Hmm... I would implement such an API rather via composition than inheritance. Such a task does not necessarily have to run in the same image but could also be accomplished on another machine. OSProcess has RemoteTask for this. Inheritance would make things complicated once one would decide to not carry out the computation as a Squeak process.
It is implemented via composition, not inheritance. Maybe it is name, ClientProcess, that made you assume it is a subclass of Process..? It is not. :)
I do like Tony's idea of process groups. We could think this one step further an add multiple tags (symbols?) to processes to support fine-granular or cross-cutting classifications. Tools like the process browser could show these tags, group by them, etc. There could be tags to identify processes that would stop working once you close the Squeak image (e.g. "not resumable")... Hmmm... Application developers could use tags to provide hints for other developers and users.
Processes already have names, which could facilitate a hierarchical naming if someone wanted, and list filtering in the Process browser would make "grouping" or "finding" a Process pretty much free.. If all you want to do is tag and group processes, I would consider putting them into named groups/containers rather than add more process-level attributes..