This Halacha is an excerpt from our Sefer
Yehalelu:  After the reading of the Haftorah Yehalelu is recited and the Torah is returned to the Aron. This is then followed by half Kaddish. [However, there were years that the Rebbe began the Kaddish immediately after the Haftorah as is normally done by Mincha of Shabbos. However in most years the Rebbe waited for the Sefer Torah to be returned.]
Aneinu: During Shemoneh Esrei of Mincha Aneinu is recited, as stated in the previous Halacha.
Nesias Kapayim: The Chazan recites Birchas Kohanim in his repetition of Shemoneh Esrei of Mincha. Those places which are not accustomed to perform Nesias Kapayim recite Elokeinu during the Chazan’s repetition. [This applies even when Mincha is being Davened early, by the time of Mincha Gedolah.] Those that are accustomed to perform Nesias Kapayim daily, are to do so as well by Mincha of a fast day. [However this only applies when Mincha is being prayed close to sunset. Some rule this to be within thirty minutes before sunset. Others rule it refers to past Plag Hamincha. Others rule it refers to Mincha Ketana. However when praying Mincha prior to Plag Hamincha Nesias Kapayim is not performed. Nevertheless one does recite Elokeinu Velokei Avoseinu, as stated above.]
Tachanun and Avinu Malkeinu: Tachanun and Avinu Malkeinu are omitted during Mincha of Erev Purim. When the fast is pushed up to Thursday, then following the Chazan’s repetition, Tachanun is recited with Avinu Malkeinu.
May Elokeinu or Nesias Kapayim be recited in the Chazan’s repetition if not everyone in the Minyan is fasting?
Some Poskim rule one is not to say Elokeinu or perform Nesias Kapayim if there are not ten people fasting within the Minyan. However, others leave this matter in question, as perhaps it follows the same ruling as the recital of Aneinu and Kerias Hatorah in which case some are lenient even if there are three people fasting.
May a Kohen perform Nesias Kapayim if he is not fasting?
No. He is thus to leave the Shul prior to Ritzei.
 Brachos 6b; Tosafus Megillah 21a; M”B 566:12 in name of Elya Raba
 M”B ibid
 Michaber 566:5
 M”A 566:7
 Michaber 5661
 Rama 566:1
The Sefaradi custom: Sefaradim do not recite the Haftorah during Mincha of a fast day. Nevertheless, if a Sefaradi is called up for Shelishi, he is to read the Haftorah with its blessings. [Yaskil Avdi 6:9; Divrei Yatziv 2:248]
 Siddur Tehilas Hashem; See Shaar Hakolel 29:4; Glosses of Rav Raskin on Siddur p. 320
 See Hiskashrus 451 p. 14 footnote 47 and 1046 p. 14; Glosses of Rav Raskin on Siddur p. 320
In the year 1988 by Taanis Esther the Rebbe began reciting Kaddish prior to the Sefer Torah entering the Heichal. However in 1989, by all the fast days [17th Tammuz, Tzom Gedalia and 10th of Teves] the Rebbe waited until the Sefer Torah was inside the Aron prior to beginning Kaddish. [Hiskashrus 1046 based on the videos of those years]. To also note that in many years that the Rebbe Davened for the Amud the Rebbe waited until after Yehalelu, and the entrance of the Sefer Torah, to the Aron to begin Kaddish, even on Shabbos by Mincha! [Hiskashrus ibid; See also Hiskashrus 1056 that this was because of a greater interval that would be caused between the end of the Kaddish and Shemoneh Esrei, and this is hence not to be followed in other Shuls in which such a long interval will not transpire.]
 See background above.
 Michaber 129:1; 566:8; Admur 129:1-2
 M”B 129:8; 566:23; Kaf Hachaim 566:52; see Admur 129:2
 Rav Poalim 4:6; So is implied from ruling of Rama and Admur 129:2 regarding saying Elokeinu by Mincha of Yom Kippur; and so is implied from Magen Avraham 129:1, as explained in Levushei Serud, as he learns the Rama. So rules Luach Davar Beito, and so is implied from Siddur.
Other Poskim: Some Poskim rule that Elokeinu is not to be said whenever Nesias Kapayim is not allowed to be done. [See M”B 129:8; Piskeiy Teshuvos 129:1 and footnote 6 which rules that it is not to be said.]
 Implication of Michaber ibid; Admur ibid; Mishneh Berurah ibid which all write that the reason why Birchas Kohanim may be done on the Mincha of a fast day is because it is customarily prayed near sunset. This thus implies that if it were to be prayed much time before sunset then Birchas Kohanim may not be done.
Other Poskim: The Chazon Ish  suggests that even when Davening Mincha early, such as by Mincha Gedolah, one may do Nesias Kapayim, although he concludes with leaving this matter in question. Based on this however, some Litvish communities have a custom to do Birchas Kohanim even when Davening early. [See Piskeiy Teshuvos 129:1; 566:8; Rav Poalim ibid]
 Kaf Hachaim 129:7; Hiskashrus 1024
 Luach Eretz Yisrael of Rav Tukichinsky; Piskeiy Teshuvos 566:8.
 Sheivet Halevi 8:23; Piskeiy Teshuvos 129:1
 Poskim ibid
 Siddur; Luach Kolel Chabad
 See P”M 566 A”A 11; Kaf Hachaim 566:52 that it may not be said if there are ten people that are not fasting.
 See Piskeiy Teshuvos 129:2; 566:8; Dvar Yehoshua 3:75
 M”B 566:14; Machazik Bracha brought in Shaareiy Teshuvah 566:3 writes 6 or 7; Machazik Bracha ibid; Yeshuos Yaakov 566; Keren Ledavid 145; Sefer Haminhagim [English] p. 26; Piskei Dinim 566; Shut Tzemach Tzedek Shaar Hamiluim 8, brought in Shaar Hakolel 9:17; Igros Kodesh 16:313
 Kaf Hachaim 129:5 in name of Peri Chadash; Machazik Bracha 129:2; Piskeiy Teshuvos 129:2