Bukhari

Hadith Commentary:
After the death of Yazid ibn Muawiyah, in 62 AH, allegiance (bay‘ah) was given to Abdullah ibn Zubair (radi Allahu anhu).
From this, some have deduced that the chapter heading of Imam Bukhari (rahimahullah) is established in this way: that the Prophet (sallallahu alayhi wa sallam) leaned on Bilal (radi Allahu anhu), from which it is understood that, when necessary, it is permissible to go to the Eid prayer while mounted (i.e., riding).
In the narration, it is also mentioned that a separate admonition was given to the women; therefore, the Imam should, after admonishing the men on Eid, also explain religious matters to the women and encourage them towards good deeds.

Hadith Commentary:
(1)
Regarding walking on foot to the Eid prayer, there is a narration from Ali radi Allahu anhu in which he said that going to the Eidgah (place of Eid prayer) on foot is Sunnah.
(Jami‘ at-Tirmidhi, al-‘Eidayn, Hadith: 530)
Imam Tirmidhi rahimahullah, in light of this hadith, has written that the majority of the scholars prefer going to Eid on foot, and they also say that no one should ride without a valid excuse.
However, in view of the chapter heading established by Imam Bukhari rahimahullah and the ahadith he presented, it appears that, according to him, both going to Eid on foot or riding are permissible. Accordingly, Hafiz Ibn Hajar rahimahullah says that perhaps Imam Bukhari rahimahullah, by establishing this chapter heading, indicated the weakness of the narrations regarding the recommendation of going to the Eid prayer on foot.
(Fath al-Bari: 2/582)
The hadith presented in reference to Tirmidhi for the recommendation of going on foot contains a narrator, Harith al-A‘war, whom Imam Nawawi has, by consensus, declared a liar. On this basis, this hadith is not valid as evidence.
(Nayl al-Awtar: 3/324, 325)
The ahadith presented by Imam Bukhari rahimahullah in this regard do not explicitly mention going on foot or riding.
This generality implies that both aspects are of equal status in this matter.
Hafiz Ibn Hajar rahimahullah has also mentioned another possibility for Imam Bukhari rahimahullah: In the narration from Jabir radi Allahu anhu, it is mentioned that when the Messenger of Allah sallallahu alayhi wa sallam addressed the women, he was holding onto the hand of Bilal radi Allahu anhu, whereas previously he had delivered the sermon without any support. When he felt tired, he took support from Bilal radi Allahu anhu. Similarly, it is better to go to the Eidgah on foot, but in case of need, riding may be used.
(Fath al-Bari: 2/582)

(2)
Should the Eid prayer be performed first or the sermon delivered first? We will discuss this later. However, there should be neither adhan (call to prayer) nor iqamah (second call) for the Eid prayer, as the narrations from Ibn Abbas radi Allahu anhu and Jabir radi Allahu anhu are clear regarding the omission of the adhan. Moreover, in some other narrations, it is also explicitly mentioned not to say the iqamah. Thus, Jabir bin Samurah radi Allahu anhu narrates: I performed the Eid prayer with the Messenger of Allah sallallahu alayhi wa sallam not once or twice, but many times, without adhan or iqamah.
(Sahih Muslim, Salat al-‘Eidayn, Hadith: 2051(887))
Although some jurists have permitted saying the words (as-salatu jami‘ah) for the Eid prayer, there is no legal permission to say such words, because in some chains of narration from Jabir radi Allahu anhu, it is mentioned that announcing the Eid prayer before it with adhan, iqamah, or any other words is not correct.
(Fath al-Bari: 2/583)
Therefore, it is not correct to announce the Eid prayer with any kind of words.
And Allah knows best.

1141. Commentary:
➊ Delivering the sermon before the Eid prayer, and calling it a speech or address, is contrary to the Sunnah.
➋ If there is a concern that the sermon’s voice may not reach the women, then it is permissible to make separate arrangements for admonition and advice for them.
➌ In an Islamic society, collecting charity and donations for religious and communal matters is not objectionable.
➍ Women may give a small amount in charity without their husbands’ permission.

1563. Commentary: The Sunnah is precisely this, because the adhan and iqamah are for the five obligatory prayers and for Jumu‘ah, as is evident from numerous ahadith. Therefore, since this was not the practice of the Messenger of Allah (sallallahu alayhi wa sallam) in the Eid prayers, not performing it is the Sunnah. And Allah knows best.