WIPING OVER THE LEATHER SOCKS (al-Mash `alal-khuffayn):
Imaam an-Nawawee -rahimahullaah- said in his explanation of Muslim (3/164):
‘Those whose sayings are counted in consensus have agreed that it is permissible to wipe over the leather socks, when on a journey and when in residence, whether it is done for a need or otherwise, to the extent that it is permissible for the woman who stays within her house, and at times when the person is not walking about. It is only criticised by the Shee`ah and the Khawaarij, and their disagreement is not counted.
al-Hasan al-Basree -rahimahullaah- said:
‘Seventy of the Companions of Allaah’s Messenger (صلّى الله عليه وسلّم ) narrated to me that Allaah’s Messenger (صلّى الله عليه وسلّم ) used to wipe upon the leather socks.’
And the best thing that is used as evidence for the permissibility of wiping is that which Muslim reported from al-A`mash: from Ibraaheem from Hammaam, who said: Jareer urinated, and then he performed wudoo·, and he wiped over his leather socks. So it was said: “You do this.” He said, “Yes, I saw Allaah’s Messenger (صلّى الله عليه وسلّم ) urinate, and then he performed the wudoo·, and then he wiped over his leather socks.”
al-A`mash said: Ibraaheem said: “This hadeeth used to please them, because the Islaam of Jareer happened after Sooratul-Maa·idah came down’’. 
an-Nawawee said  : ‘Meaning that Allaah- the Most High- said:
“Then wash your faces and your arms up to and including the elbows, and wipe your heads, and wash your feet up to and including the ankle bones.” [Sooratul Maa·idah (5):6]
He said: “So if the Islaam of Jareer had come before the descent of al-Maa·idah it would have been possible that his hadeeth about wiping over the leather socks was abrogated by the Aayah of al-Maa.idah, but since his Islaam came later on, then we know for certain that his hadeeth is to be acted upon; and it clarifies that the one who is meant by the Aayah is the person who is not wearing leather socks. So therefore the Sunnah particularises the Aayah, and Allaah knows best.’’
It is a condition for the permissibility of wiping that he should wear the leather socks upon the state of having made wudoo·.
From al-Mugheerah ibn Shu`bah -radiyallaahu `anhu- who said: I was with the Prophet (صلّى الله عليه وسلّم ) one night on a journey. So I poured water out for him from the pot, and he washed his face and his two forearms, and he wiped his head. Then I reached down to remove his leather socks, so he said: “Leave them because I entered them both in a state of purification.” And he wiped over them. 
The time period for wiping:
From `Alee ibn Abee Taalib -radiyallaahu `anhu- who said Allaah’s Messenger (صلّى الله عليه وسلّم ) laid down three days and their nights for the traveller,and one day and night for the resident. 
The place of wiping and its description:
The place which it is legislated to wipe is the upper surface of the leather sock, because of the saying of `Alee ibn Abee Taalib -radiyallaahu `anhu- : “If the Religion were in accordance with opinion, then underneath of the leather sock would have more right to be wiped then its top part, but I saw Allaah’s Messenger (صلّى الله عليه وسلّم ) wiping over the upper surface of his leather socks.” 
And what is obligatory in wiping is whatever the term ‘wiping’ is applied to.
Wiping over socks and sandals:
And just as it is permissible to wipe upon leather socks then it is permissible to wipe upon normal socks and upon sandals, because of the hadeeth of al-Mugheerah ibn Shu`bah that the Prophet (صلّى الله عليه وسلّم ) performed the wudoo·, and he wiped over the socks and the sandals. 
And from `Ubayd ibn Juraij who said: “ It was said to Ibn `Umar: We have seen you doing something which we did not see anyone else besides you doing.’ He said: ‘And what is it?’ They said: ‘We have seen you wearing these animal- hair sandals.’ He said: ‘I saw Allaah’s Messenger (صلّى الله عليه وسلّم ) wearing them and performing wudoo· in them, and he wiped over them.’ 
That which nullifies the wiping:
The wiping is annulled by one of these three:
1) Completion of the time period, because wiping has a time limit, as you have known. So it is not permissible to increase upon the time period which has been established.
2) Janaabah: Because of the hadeeth of Safwaan: “Allaah’s Messenger (صلّى الله عليه وسلّم ) used to command us when we were upon a journey that we should not remove our leather socks for three days along with their nights except from janaabah, but rather from passing excrement, and urine and sleep.” 
3) Removing that which has been wiped upon from the feet, because if he removes them, and then wears them again, then he will not have entered his two feet in a state of purification.
A point of benefit:
The completion of the time period, and removing that which he has wiped upon nullify the wiping alone. So it is not permissible to wipe until he has performed the wudoo·, and washed his feet, and then worn it. However if he was still in a state of wudoo· when he took off the thing that he wiped over, or when the time period finished, then he remains upon his state of wudoo·. He can pray with it as he wishes until he breaks his wudoo·.
A point of benefit:
Whoever wears two pairs of socks upon a state of purification, and then wipes over them, and then he removes the top one -after having wiped over it, then it is permissible for him to complete the time period by wiping over the underneath one, because it is correct that he entered his two feet into them in a state of being pure. But if he wore a sock, and wiped over it, and then he wore another one on top of it, then he may not wipe over that. This is because it will not be correct that he has entered the two of them when they were pure. 
 Saheeh: [Mukhtasar Muslim (136), Muslim (1/227/272), at-Tirmidhee (1/63/93).
 Sharh Muslim (3/164)
 Agreed upon: Muslim (1/230-79-274), al-Bukhaaree (1/309/206)-in abridged form, Aboo Daawood (1/256/151)
 Saheeh:[ Mukhtasar Muslim (139), Muslim (1/232/276), an-Nasaa·ee (1/84).
 Saheeh: [Irwaa.ul-Ghaleel 103], Aboo Daawood (1/278/162).
 Saheeh: [Irwaa·ul-Ghaleel 101], Aboo Daawood (1/269/159), at-Tirmidhee (1/67/99), Ibn Maajah (1/185/559).
 Ibn Khuzaymah (1/p.100 hadeeth 199), al-Bayhaqee (1/ p.287 )
 Hasan: [Irwaa·ul-Ghaleel 104], at-Tirmidhee (1/65/96), an-Nasaa·ee (1/84)
 This is what Shaikh al-Albaanee mentioned to me.
Source : Al-Wajeez, The Book Of Purification – Abdul-`Azeem ibn Badawee al-Khalafee, Translated by Abu Talhah Dawood Burbank . An excellent book on the rules and regulations of various acts of worship, with evidences from the Book and Sunnah. This chapter deals with water and aspects of purification, wudhoo, ghusl, the toilet etc
Related Link: https://abdurrahman.org/tahara-purification/