I haven't flown international on DL/ Skyteam since 2022 so I don't know how much rules have changed.

When arriving on a KE flight in Prestige class, connecting to a domestic DL First flight, will I get SkyClub access? Will it matter if the KE flight is KE-issued, or DL code-share?

The rules on the SkyClub page mentions "same-day international business" on a partner without explicitly mentioning if connections and arrivals are covered.

Last time I flew KE/DL in 2022, I was Gold so that may have got me in. I am just a Silver now with DL.