Although running on Shabbos is generally not allowed, here it is permitted since it is considered as if he is running for the sake of a mitzvah; O.C. 301:1.
Rav S. Z. Auerbach (Shemiras Shabbos Kehilchasah18, footnote 245). In this case, it is preferable to let the item drop out of the pocket in a backhanded manner; Orchos Shabbos28:71.
Therefore, this option cannot be used if going back to the point of departure (or to an enclosed area) would require stopping along the way for any reason, such as waiting for a traffic light.
See Mishnah Berurah266:36.
Sha’ar Hatziyun349:18.
This is to satisfy the first view quoted by Rema345:19. See Beiur Halachah (ה"דשיו) who holds that an exempt area in a karmelis must be over 10 tefachim tall, as per O.C.345:18.
Because, otherwise, it would be forbidden to walk four amos in a public domain (or a karmelis) while carrying the item – even if the hanachah will be in an exempt area; see Kitzur Hilchos Shabbos (Posen) 39:16.
See Nefesh Chayah5; Eretz Tzvi, 1:76; Igros Moshe2:112 (and Mesores Moshe, vol. 2, pg. 63); Nesivos Shabbos9, footnote 35; Shemiras Shabbos Kehilchasah18:59.
