For those who arrive at the synagogue late, after the beginning of Birkot Keriat Shema, and miss the response to Barchu by the congregation, the Chachamim established that after the prayer service, the chazan repeats Barchu. The latecomers, along with the entire congregation, answer, “Baruch Hashem HaMevorach l’olam va’ed” (“Blessed is Hashem, the Blessed One, for all eternity”). That is also how we practice at the conclusion of the Ma’ariv service. According to this, on Shabbat and holidays, there is no need for the chazan to say Barchu at the end of the prayer service, because it is reasonable to assume that even those who came late succeeded in hearing the recital of Barchu by the people called up to the Torah (Shulchan Aruch 133:1; Maharitz; Mishnah Berurah, the introduction to section 69). For this reason, those who pray in Nusach Ashkenaz do not recite Barchu after the prayer service on those weekdays on which the Torah is read. However, on days when the Torah is not read, it is always customary to recite Barchu without verifying whether there is someone who needs to make it up, so as not to trouble the congregation to determine if there is a latecomer present.
According to the Ari, Barchu is always recited after the prayer service, because according to his Kavanot, it is necessary to say Barchu twice in every prayer service, once before Birkot Keriat Shema and a second time at the end of the service. The same is true for Ma’ariv. That is the custom of those who pray in Nusach Sephard, and the minhag of the Chassidim as well (Kaf HaChaim 133:1).
In every nusach of prayer, Barchu is recited after Kaddish d’Rabbanan, which is the last Kaddish, so that even the last of the latecomers will succeed in hearing it. It is customary that the one who recites Kaddish says the Barchu as well. However, sometimes the one reciting the last Kaddish is a child who lost a parent and has not yet reached the age of mitzvot. In such a case, the chazan, who is obligated to perform the mitzvot, must be the one to recite Barchu (Mishnah Berurah 55:4).