since Jest doesn't support ES5, I guess we could have - for now - Jest to connect to an ES2 cluster, and a different one for ES5+

I suspect it would be more work than just dropping Jest and moving to our own client based on the official low-level REST client, but I'll keep it in mind as an option.

hopefully we won't need to duplicate too many tests?

No, but those we need to duplicate are big ones. A shame, but yes, duplication seems like the only workable solution.

we could pick a "favourite" version which is going to run all tests locally by default, and then have CI report on combinations.

Ok, let's do that. Maybe it would be the right time to introduce Jenkins pipeline-as-code? I'll try it once I'm done with the rest.

I think the main think to decide is how to implement those "ES dialects", like the JIRA title suggests. My vote goes to use the driver which the ES team is maintaining, I'm confident they would be responsive on our feedback as we make progress.

It's also my opinion that we should use the official, low-level REST client. I'll try that.