A light aircraft pilot entering the pattern to a non towered airport reported a NMAC with a departing aircraft which turned in front of them.

Narrative

I was flying a cross country flight from ZZZ1 to ZZZ. Radar services were terminated by ATC approximately over the bridge. I continued VFR to ZZZ; and make two radio calls to the ZZZ CTAF with a position report and intentions to enter the left downwind Runway XX on the 45 for a full stop landing. A Piper Cherokee took off Runway XX for a left downwind departure in front of me while still on 2nm '45 to the downwind' despite my position and intention reports. As I approached the downwind the Cherokee was at my 10 or 11 o clock on a roughly crosswind heading at the same altitude. It seemed to have been a mix of the Cherokee turning a premature crosswind to avoid noise sensitive areas south of the airport; and myself entering the downwind slightly downwind of midfield. I got the Cherokee in sight although not early enough to avoid taking evasive action. I turned right to the downwind and began a descent to 600ft AGL and flew the pattern at that altitude. The Cherokee passed right above me with between 400-500ft of vertical separation. I landed without further complications. Although we both eventually saw each other and confirmed as much; I believe it warrented a report. The Cherokee was operating without ADS-B out.