The honest answer is most teams don't manage this well and end up burning prospects with overlapping sends. I've seen this mess up entire campaigns.
Your best bet is centralizing suppression management in one place. HubSpot can be your source of truth since it's already your CRM. Create global suppression lists there and sync them out to your other platforms via API or manual exports.
For Instantly, you'll need to regularly export your unsubscribes and bounces back to HubSpot. Same with Beehiiv — their suppression data needs to flow back. Luma is trickier since their API is limited, but you can at least manually sync key segments.
The nuclear option is using a tool like SendGrid or Postmark as your email infrastructure backbone and routing everything through there. Gives you unified suppression and frequency capping, but requires more dev work.
Quick tactical fix: Set up weekly suppression syncs. Export unsubscribes from each platform and import to the others. It's manual but prevents the worst overlaps.
Honestly though, having four email platforms is creating unnecessary complexity. Most teams I know consolidate to 2-3 max — one for marketing automation (HubSpot), one for outbound (Instantly/Smartlead), and maybe one for newsletters. The operational overhead of managing more usually isn't worth it unless you have dedicated ops resources.
Consider which platforms are actually driving results and consolidate where possible.