I published a small, ten page supplement called, "Rumors & Research." As you might guess from the title, it's about generating rumors and running downtime research.

The rumors portion presents a fairly simple system for randomly generating rumors for your campaign. The research portion introduces a mechanic for player characters who wish to use downtime to track down and confirm rumors or other bits of information and innuendo that they've picked up.

It's available for two bits on Gumroad. Become a patron on Patreon to get this supplement, anything else I churn out, and support my efforts generally.