Feature request: Sync conversations and sessions between desktop and mobile (Dispatch)

Summary

As a Cowork user, I'd like to access my desktop Claude conversations and sessions from the mobile Dispatch interface.

Use case

I often work on programming and other tasks from my PC during the day. Later, when I'm away from the computer, I'd like to check in on what we were working on, ask follow-up questions, or pick up the thread from my phone through Dispatch.

Right now each conversation is siloed to the device where it started. Dispatch on mobile begins every session blank and has no visibility into desktop conversations or Claude Code sessions running on the PC.

Suggested behavior

• Mobile Dispatch shows a list of recent desktop conversations (regular chats and Claude Code sessions).
• Tapping one lets me read the transcript.
• Optionally: continue the conversation from mobile, syncing back to the desktop session.

Why it matters

The whole point of mobile access is continuity. Without shared history between devices, Dispatch on the phone is a separate tool rather than an extension of the work I'm already doing on my PC.

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Note: I'm aware this repo is for the Claude Code CLI and the request is really about Cowork/Dispatch — filing here anyway, knowing it may be redirected.