Naming, Resuming, and the Right Execution Mode for a Multi-Stage Subagent Task

You dispatch a subagent named db-migrator to plan and start a schema migration. Partway through, you want to (a) let it keep working while you review something else in the main conversation, and (b) later send it a follow-up instruction that continues its work with full context of what it already did, rather than starting a new subagent from scratch.

1. What determines whether db-migrator runs in the foreground (blocking your session) or the background, and what tradeoff does each mode carry for tool access and when you see its permission prompts?
2. How do you address db-migrator specifically for that follow-up instruction, mechanically - what identifiers can be used, and why does this matter more than just saying "continue" in a new prompt?
3. A teammate suggests getting the same "resume with context" effect by spawning a brand-new subagent and pasting in a copy of the transcript so far, instead of addressing the existing subagent directly. What's wrong with that as a substitute?