Praise be to Allah.
It is permissible for a person to wipe over the khuffs of another person, as it is permissible for him to help him to do wudu’.
Ibn ‘Abidin (may Allah have mercy on him) said in his commentary on al-Bahr al-Ra’iq (1/182): If a person tells someone else to wipe over his khuffs and he does that for him, it is valid, as it says in al-Khulasah. End quote.
It is not stipulated that the wiping should be done directly with the hand. So if the wiping is done with a cloth or a wetted broom – as you mentioned – it is permissible.
An-Nawawi (may Allah have mercy on him) said in al-Majmu‘ (1/549): Our companions said: It is valid to wipe with the hand, or with a finger, or a piece of wood, or a piece of cloth, or something else. End quote.
This is the view of the Hanbalis. Please see: al-Insaf (1/160, 1/185).
It is valid to sprinkle water on the khuffs with a spray bottle and the like.
Zakariyya al-Ansari said: If he puts his wet hand on the khuff without passing the hand over it, or he drips water onto it, that is valid.
End quote from Sharh Manhaj at-Tullab ma‘a Hashiyat al-Bujayrami (1/88).
If the shoes cover the ankles and were put on after purifying oneself completely, it is permissible to wipe over them, because that comes under the same ruling as khuffs. But if the shoes do not cover the ankles, then it is a matter concerning which there is a difference of scholarly opinion as to whether it is permissible to wipe over them.
The majority are of the view that it is not permissible to wipe over such shoes. However some scholars – such as al-Awza‘i, Ibn Hazm and Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) – were of the view that it is permissible to wipe over them. Shaykh al-Islam limited that to sandals that are difficult to take off except by using a hand or foot.
The evidence which indicates that it is permissible to wipe over sandals is as follows:
The report narrated by al-Bazzar, according to which Ibn ‘Umar (may Allah be pleased with him) did wudu’ with his sandals on his feet; he wiped over them and said: This is what the Messenger of Allah (blessings and peace of Allah be upon him) used to do.
Its isnad was classed sahih by Ibn Hajar in ad-Dirayah fi Takhrij Ahadith al-Hidayah (1/83).
The report narrated by Ibn Dawud (16) from Aws ibn Abi Aws ath-Thaqafi, which says that the Messenger of Allah (blessings and peace of Allah be upon him) did wudu’ and then wiped over his sandals and feet. Classed as sahih by al-Albani in Sahih Abi Dawud.
The report narrated by Ibn Abi Shaybah from Abu’-z-Zabyan, who said: I saw ‘Ali urinate whilst standing, then he did wudu’ then wiped over his sandals, then the mu’adhdhin gave the call to prayer, so he took them off.
It says in al-Furu‘ (1/160): Our shaykh [i.e., Ibn Taymiyah] favoured the view that it is permissible to wipe the foot and sandals that are difficult to remove except by using a hand or foot, as is mentioned in the reports. He said: It is sufficient in this case to wash or wipe most of the foot itself, or what is apparent of it, and that is more appropriate than wiping part of the khuff. End quote.
See: al-Muhalla (1/336) and al-Insaf (1/179).
As for the majority of scholars, some of them regarded these hadiths as da‘if (weak), and some of them interpreted them as referring to wiping over sandals that are worn over socks.
According to this view, if the sandals were put on over socks, it is permissible to wipe over both of them, because of the report narrated by at-Tirmidhi (99), an-Nasa’i (125), Abu Dawud (159) and Ibn Majah (559) from al-Mughirah ibn Shu‘bah, who said: The Prophet (blessings and peace of Allah be upon him) did wudu’ wiping over his socks and sandals. This hadith was classed as sahih by al-Albani in Sahih Abi Dawud.
Ibn al-Qayyim (may Allah have mercy on him) said in his commentary on Sunan Abi Dawud: What appears to be the case is that he wiped over socks over which sandals were worn. This is what is to be understood from the report, because they are mentioned separately… Moreover, what was narrated from ‘Umar ibn al-Khattab concerning that is that he wiped over the straps of the sandals that went over the top of the foot, along with the socks. As for the bottom part of the foot and heel, he did not wipe over them. End quote.
It was narrated in ‘Awn al-Ma‘bud that at-Tahawi said: He wiped over sandals under which there were socks, and his aim in doing so was to wipe over the socks, not the sandals. End quote.
In Masa’il Salih ibn al-Imam Ahmad (379) it says: I asked him about wiping over the sandals. He said: If there are socks on the feet that are tight on the feet, then there is nothing wrong with wiping over the sandals. End quote.
To sum up: according to the first view – which is the view of al-Awza ‘i and those who agreed with him – it is permissible to wipe over shoes that come below the ankles, whether socks are worn under them or not.
According to the view of the majority, if the person put on socks (that cover the ankles) when in a state of purity, and he put shoes on over them, then it is permissible to wipe over both of them.
When it is valid to wipe over the khuffs, it is valid to sprinkle water on them, as noted above, or to pour water on them, and in that way he will wash the khuffs. However, some jurists regard it as makruh (disliked), some of them said that it is only disliked if the water will damage them, and some of them stipulated that the individual should also wipe [the khuffs] as well as washing them.
It says in Kashshaf al-Qina‘ (1/119): The ruling on washing [the khuffs] is like the ruling on wiping over the head, as discussed above; it is valid to wipe it in addition to washing it.
It is makruh (disliked) to wash the khuffs, because that will damage them. End quote.
It says in Hashiyat al-Jamal ‘ala Sharh al-Manhaj (1/146): It is disliked to wash the khuffs repeatedly, because that will damage them.
From this we may learn that if the khuffs are made of iron and the like, it is not disliked to wipe or wash them repeatedly, because that will not damage them. And this is the same. End quote.
So he should limit it to sprinkling over the khuffs or dripping water onto them, and not pour water on them. If he uses a broom and the like, that will be better.
And Allah knows best