The majority of Geonim and Rishonim maintain that there is no need to recite the passages of the Korbanot before praying Minchah. Neither the Rambam (Hilchot Tefillah 9:8), nor the Shulchan Aruch mention that Korbanot are recited at Minchah. Still, there are Rishonim who write that the passages of the Tamid and the Pitum HaKetoret are recited before Minchah, just as it is proper to say them before Shacharit (Rama 234:1; 132:2). Those who follow the Ari are more meticulous about reciting Korbanot before Minchah (Kaf HaChaim 232:6; Piskei Teshuvot 234:1).
All are accustomed to saying Tehillah L’David (Ashrei) before the Amidah of Minchah because it is proper to praise Hashem prior to praying the Amidah. It is, indeed, a kind of Pesukei d’Zimrah before the Minchah prayer (see earlier in this book 14:1-2). However, it is not an absolute obligation to recite it before Minchah, and therefore, in a situation in which the time to pray Minchah is about to lapse, one must skip Tehillah L’David and immediately start reciting the Amidah (Mishnah Berurah 234:6). Likewise, if the congregation is ready to begin the Amidah, he should skip Tehillah L’David so that he can start praying together with them in a minyan (Mishnah Berurah 108:14; Or L’Tzion, part 2, 15:3).
After Tehillah L’David the chazan recites Half-Kaddish, following which the congregations begins the silent Amidah. Subsequently, the chazan says Chazarat HaShatz. Birkat Kohanim is not recited in Chazarat HaShatz of Minchah, for fear that perhaps the Kohanim drank wine at the meal beforehand and will come to bless the nation when they are inebriated. However, on a public fast day, there is no such concern, and therefore they bless the nation then (on condition that the congregation prays after plag haminchah, as explained earlier in this book 20:5).
After the Amidah, prayers of supplication (Tachanunim) are recited. According to the Sephardic custom, Vidui, the Thirteen Attributes of Mercy, and Nefillat Apayim are all said; whereas according to the Ashkenazic and Yemenite (Baladi) minhag, only Nefillat Apayim is recited. Subsequently, the chazan says Kaddish-Titkabal. On days on which Tachanun is not recited (as explained earlier in this book 21:7-8), the chazan recites Kaddish-Titkabal immediately following the Amidah repetition.
Sephardim say “Lamenatze’ach Bin’ginot” (Psalm 67) followed by Mourner’s Kaddish, and conclude the prayer with Aleinu L’Shabe’ach. Ashkenazim do not recite Lamenatze’ach. They say Aleinu L’Shabe’ach and then Mourner’s Kaddish.