A Mooney pilot reported the engine was sputtering due to loss of fuel due to loss of fuel cap in flight and then quit while taxiing on the ramp.

Narrative

We diverted from ZZZ1 to ZZZ for Customs on a return flight from ZZZZ with 5 1/2 hours fuel on board. Approaching ZZZ after 3 hours total flight time; we experienced fuel starvation in the right tank with a sputtering engine. Switching to the left tank; the engine caught but I noticed the fuel cap missing on the left tank. As we got closer the Tower Controller requested we deviate to the west to sequence behind a jet. I realized as the engine would sputter and regain power repeatedly; that we did not have fuel left in the left tank to comply. I [requested priority handling] and was immediately directed to the Runway XXR. After landing the engine stopped sputtering and we taxied to Customs where the engine died before I could turn it off. There were no injuries or aircraft damage. We fueled the right tank and flew to our home airport ZZZ2.While I checked the fuel and secured in the right tank after the local FBO filled the plane before departure from ZZZZ; I failed to check the left tank fuel and secure the cap on the left tank. Both tank sumps were drained prior to departure. We don't know when the left fuel cap departed the plane.