A relatively similar case to those you ask about is what is called poress al Shema. This refers to a case in which people who do not have a minyan for all of davening want to join with others, including those who have already finished davening, so as to enable the former group to recite Kaddish (following and relating to P’sukei D’Zimra) and Barchu. The Shulchan Aruch[2] rules that although it is best if there are six men present who have yet to take part in these devarim shebekedusha,[3] Kaddish and Barchu may be said even for the needs of only one such person. Furthermore, if need be, even someone who already took part in a minyan may lead the recitation.[4]
Similarly, the Taz[5] infers from several sources that in order to say Kaddish, which relates to a previous text(s) that was recited, it suffices for the minyan to be assembled at the time the Kaddish is recited, even if the members of the minyan were not present for the matters to which the Kaddish relates. The Magen Avraham, on the other hand, rules regarding the Kaddish after learning Torah[6] and the one after Ashrei at Mincha[7] that the minyan must already have been present during the previous recitations to which the Kaddish relates. The Pri Megadim[8] posits that in the case of the Kaddish following P’sukei D’Zimra, the Magen Avraham would agree with the Taz’s leniency, because P’sukei D’Zimra/Kaddish is a requirement. In other words, in cases such as poress al Shema, when there is a real obligation for the recitations to be done, the minyan enables the recitation of Kaddish whenever it becomes possible. In contrast, regarding less critical recitations, including Ashrei at Mincha, the need for Kaddish is created only when a minyan is present during the previous recitation.
Some maintain that even in cases in which the Magen Avraham’s opinion applies, he only requires the ten to be present when the texts that justify the Kaddish afterward are recited; he does not require that all ten men actually participate in the learning/recitation.[9] On the other hand, the Aruch HaShulchan[10] is fully stringent regarding Kaddish after learning, which is optional, requiring that ten people must have learned together.[11]
The best way to avoid any machloket in cases of Kaddish after learning is for the learners to include the rest of the quorum in a “Kaddish-trigger” statement, such as the famous statement of Rabbi Chananya ben Akashya. This works in a variety of scenarios, because even if the rest of the recitation did not count, the trigger-statement is itself grounds for Kaddish. If there are fewer than ten men reciting Pitum HaKetoret at the end of davening, it is worthwhile for the chazan to say, “Rabbi Elazar…: Talmidei chachamim…” or the last three p’sukim out loud. If this is difficult, one can rely on the majority opinion that we do not require even a majority of the minyan to be learning before Kaddish D’Rabbanan.[12] Furthermore, the Aruch HaShulchan probably agrees that the presence of ten men is enough for Pitum HaKetoret, because it is a set part of tefilla for those communities who recite it regularly and/or because having at least one Kaddish D’Rabbanan at Shacharit is important.[13]
Regarding the question of whether someone not involved in the part of tefilla to which the Kaddish is related may nevertheless recite it, we saw that this is allowed in the case of poress al Shema. The Mishna Berura[14] assumes that this is also true for Kaddish after learning, and there is no reason to assume this would not apply to the Kaddish after Aleinu, Shir Shel Yom, or Pitum HaKetoret.[15] However, Rav Moshe Feinstein rules that a non-participant is permitted to recite the Kaddish only if the matter to which the Kaddish relates is part of the tefilla or is part of a learning session of the tzibbur, such as Pitum HaKetoret. It is not permitted for one who did not take part in a free-standing learning session that just happened to include ten men to recite the Kaddish after it.[16] Even in the latter case, however, it is enough to have heard and understood only a small part of the learning, and the passage of Rabbi Elazar suffices for this.[17]