I’m using Cloudflare with full site caching (HTML caching) and server push is sometimes working/not working depending on the first request and load of the page. Requesting the page in some clients it cause the server push headers gone from the response.
I checked it in both Chrome Dev-tools, GTmetrix, etc and sometimes cf-h2-pushed header appears as it should, containing all the assets, but sometimes not.
It occurs especially if I load a page with Google PageSpeed Insight. Maybe PageSpeed Insights is a client that doesn’t support server push, so then Cloudflare miss it from the cache?