This is a good chain of transmission, indicating that it has an established basis from Asmāʾ. Perhaps Ibn Ḥarmalah had two chains from him: at times he narrated it from Yaḥyā ibn Hind, and at other times from Ibn al-Musayyab. And Allah knows best.
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As for the hadith of Abū Hurayrah, it is narrated by Ḥabīb ibn ʿAbd Allāh, from Shubayl, from him. He said: The Prophet, peace and blessings be upon him, was fasting on the day of ʿĀshūrāʾ and said to his Companions: “Whoever among you began the morning fasting should complete his fast, and whoever has eaten his household’s breakfast should complete the rest of his day.” Aḥmad recorded it (2/359). Its narrators have been deemed trustworthy, except for Ḥabīb ibn ʿAbd Allāh al-Azdī al-Yaḥmadī, who is unknown.
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As for the hadith of Ibn ʿAbbās, it is narrated by Jābir, from ʿIkrimah, from him. He said: “The Messenger of Allah, peace and blessings be upon him, sent word to the people of a village four farsakhs away—or, he said, two farsakhs—on the day of ʿĀshūrāʾ, and instructed whoever had eaten not to eat for the remainder of the day, and whoever had not eaten to complete his fast.” Aḥmad recorded it (1/232). Jābir is Ibn Yazīd al-Juʿfī, and he is weak.
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As for the hadith of the men of Aslam, Ibn Shihāb narrates it from Ibn Sindar, from men among them [from Aslam] (1), that the Messenger of Allah, peace and blessings be upon him, said to a man from Aslam: … then he cited it in wording similar to the first hadith. Al-Nasāʾī recorded it in al-Kubrā (38/1): Aḥmad ibn Ibrāhīm informed us; he said: Yazīd—meaning Ibn Mawhib—narrated to us; he said: al-Layth narrated to me from ʿUqayl, from Ibn Shihāb. This is a chain whose narrators are trustworthy except for Ibn Sindar. Al-Ḥāfiẓ stated definitively in the chapter, “Those Who…”