Kohen
1. To wash his hands. Some say if a Kohen would need to walk within four amos in front of someone davening Shemoneh Esrei to wash his hands, it is better to rely on the opinion that a Kohen can rely on the netilas yodayim he did before davening if he did not lose focus of it [היסח הדעת] and did not touch an unclean area (הגר''נ קרליץ). However, most poskim agree a Kohen may walk in front of someone davening because the washing is for a mitzva purpose (הגרי''ש אלישיב, הגר''ש וואזנר קובץ מבית לוי ד'; see Issues 311 and 312 at length).
2. To go up for Birkas Kohanim. If there is no other path, a Kohen may walk in front of someone davening to get to the front for Birkas Kohanim, as it is a passing mitzva [מצוה עוברת] and a mitzva involving many people [מצוה דרבים]. If there are two Kohanim, it is also a mitzva d’oraisa (הגר''ח קנייבסקי בשם החזו''א).
Levi
3. To wash a Kohen’s hands. A Levi should not walk in front of someone davening to wash a Kohen’s hands, as that is not essential to the mitzva (ד' אמות של תפלה ח''ב סי' ז' הע' 84).
Yisroel
4. To stand in front of the Kohen. Someone by the front wall who wants to back up to stand in front of the Kohen (שבילי דוד סי' ק''ב) should not walk in front of someone davening for this purpose. The same applies to those who come from all parts of the shul to stand directly in front of the Kohen during the bracha. This is because doing this is just a minhag, not an actual mitzva.
