Definition from Wiktionary, the free dictionary
See also: run in
For the meaning "end-phase", it presumably comes a marathon race, where in the final part the runners run into the stadium and complete a lap.
Audio (US) (file)
- An encounter; a scrape or brush, especially one involving trouble or difficulty.
- I heard his brother had a run-in with the police the other day.
- end-phase of a competition