Restore the work context
Name the team, project, period, or challenge you shared and add a genuine reason they came to mind. A short specific message is easier to trust than broad networking flattery.
Be transparent if there is a request
If you want advice, a referral, information, or an introduction, say so and include what it requires. Do not disguise a favor request as a social catch-up.
Respect professional boundaries
Use contact information they made available for professional communication. Do not ask for confidential company data or pressure them to recommend work they cannot endorse.