The Rema in Shulchan Aruch, Orach Chaim 581 notes how the person who leads the Selichos continues with davening. He then notes an additional custom:
ויש מקומות נוהגים שהמתפלל סליחות יתפלל כל היום
And there are places where the one who davens selichos, davens all the day.
The Ba'er Heitev explains that this is because, "שהמתחיל במצוה אומרים לו גמור" - "The one who begins with a mitzvah, we say about him that he completes it", and this is why the one who lead Selichos takes precedence over even someone who is an avel (mourner), mohel (someone who performs a circumcision) and someone who is commemorating a Yahrzeit (the anniversary of a relative's death).
The Midrash Tanchuma on Parshas Eikev explains the reasoning because:
Rabbi Yochanan said, "Anyone who began with a commandment, and afterwards another comes and finishes it, it will be called according to the name of the one who finishes it."
Do we have any other examples in Halacha where we apply this dictum?