Ibn Majah

Commentary:
(1)
The reason for the ruling regarding rinsing the mouth (kuli) that has been mentioned indicates that its purpose is the cleanliness of the mouth, and it is not related to the validity or invalidity of ablution (wudu).

(2)
Cleanliness holds great importance in Islam; therefore, rinsing the mouth (kuli) and using the tooth-stick (miswak) have been legislated in ablution (wudu) as well. The residue of greasiness remaining in the mouth after eating or drinking is contrary to the principles of hygiene; therefore, after drinking milk or consuming any oily food, special care should be taken to clean the mouth.

Hadith Commentary:
➊
Whether milk is fresh or warm, it contains fat.
After its consumption, the tongue also becomes coated with greasiness, due to which recitation of the Qur’an becomes somewhat difficult, so one should rinse the mouth after consuming milk.
In some narrations, there is an instruction to perform ablution (wudu) after drinking milk.
(Sunan Abi Dawud, Book of Purification, Hadith: 498)
This matter is not for obligation but rather for recommendation, because the narrator of this report, Ibn Abbas (radi Allahu anhuma), says that if I do not rinse my mouth after drinking milk, I do not mind.
In addition, Anas (radi Allahu ta'ala anhu) reports that the Messenger of Allah (sallallahu alayhi wa sallam) drank milk, and after that neither rinsed his mouth nor performed ablution, rather, after that he performed prayer.
(Sunan Abi Dawud, Book of Purification, Hadith: 197)
Some have considered the hadith of Anas as abrogating (nasikh) the hadith of Ibn Abbas (radi Allahu anhuma), but this position is not correct, because no one holds that rinsing the mouth is obligatory such that there would be an issue of abrogation.
(Fath al-Bari: 1/409)
The point is that if there is a need to pray immediately after drinking milk, one should rinse the mouth; if after some time the mouth becomes clean due to saliva or on its own, then rinsing is not necessary, because the purpose of rinsing is to remove the greasiness, and if it is removed by saliva itself, then rinsing is not required.

➋
Imam Bukhari rahimahullah has mentioned the issue of corroboration (mutaba‘at) at the end of the hadith.
The corroboration of Yunus ibn Yazid is mentioned in Sahih Muslim (Hadith 358) in a connected (mawsul) form, while the narration of Salih ibn Kaysan has been reported with a chain by Abu al-Abbas al-Sarraj in his Musnad.
In addition, Imam Awza‘i rahimahullah has also corroborated ‘Uqayl, which Imam Bukhari rahimahullah himself has mentioned in the Book of Foods.
(Fath al-Bari: 1/409)

1:
Some people have declared it obligatory.
Their evidence is that in a narration of Ibn Majah (no. 498), ((مَضْمَضُوا مِنَ الِّلبَنِ)) appears in the imperative form, and the original ruling regarding an imperative is obligation.
The response to this is that this applies only when there is no evidence to interpret it as recommended (mustahabb), and here there is evidence to interpret it as recommended, because Abu Dawud (no. 196) has transmitted a narration from Anas that the Prophet sallallahu alayhi wa sallam drank milk and neither rinsed his mouth nor performed ablution (wudu).
Its chain of narration is hasan (good).

Benefits and Issues:
With regard to such foods and drinks that contain fat, it is preferable and superior to rinse the mouth so that the mouth remains thoroughly clean during prayer. In the upcoming hadith, the concession regarding this will be mentioned.

187. Commentary: The effects of milk, especially its greasiness and sweetness, remain in the mouth; therefore, rinsing the mouth (making mouthful rinses) after drinking milk is recommended (mustahabb).

Benefits:

Imam Nawawi rahimahullah says: The hadiths are evidence that rinsing the mouth (madmadah) after drinking milk is a recommended (mustahabb) act. The scholars state that similarly, rinsing the mouth after consuming any food or drink is also a recommended action, so that nothing remains stuck in the teeth or mouth which one might chew during prayer. Furthermore, rinsing the mouth is recommended so that the greasiness or residue from eating and drinking is removed and the mouth is cleansed. [نووي: 45/4]