Wa `alaykum As-Salamu waRahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
1- If the shoe covers the foot with the ankles, then it is permissible to wipe over it.
2- But if it does not cover the area that must be washed, which is the entire foot with the ankle, then it is not permissible to wipe over the shoes.
3- If a person wipes over the shoes that cover the area that must be washed, then he takes them off when he is still in a state of purity, his wudu is not rendered invalid.
Answering your question, the Fatwa Center at Islam Q and A, states:
If the shoe covers the foot with the ankles, then it is permissible to wipe over it, because it is like the khuff [a type of leather sock that covers the foot and ankle; the word may also refer to a boot or any footwear that also covers the ankle].
But if it does not cover the area that must be washed, which is the entire foot with the ankle, then it is not permissible to wipe over the shoes according to the majority of jurists. (Al-Mawsu`h al-Fiqhiyyah al-Kuwaitiyyah, 27:264)
This is the view favoured by Sheikh Ibn Baz and the Standing Committee for Issuing Fatwas.
Sheikh Ibn Baz (may Allah have mercy on him) said: “One of the conditions of it being permissible to wipe over the khuffayn and the socks is that they should cover the area that must be washed.” (Majmua al-Fataawa by Ibn Baz, 10:111, Fatawa al-Lajnah ad-Da’imah, 5:396)
If a person wipes over the shoes that cover the area that must be washed, then he takes them off when he is still in a state of purity, his wudu is not rendered invalid, according to the correct scholarly view.
But it should be noted that the concession allowing him to wipe over them ends when he takes them off, so if he puts them on again and wants to do wudu, he has to take off his shoes and socks and wash his feet.
If he puts on socks and puts over them shoes that do not cover the ankles, then there are three scenarios:
1- He wipes over the shoe only; we have stated above that this is not permissible.
2- He wipes over the socks only, in the sense that he takes off his shoes and wipes over his socks with his hands, then put the shoes back on.
This is permissible and there is nothing wrong with it, and in this case it is permissible for him to take off the shoes, and that does not render his wudu invalid.
3- He wipes over the shoes and socks together; this is also permissible
If he wipes over the shoe that does not cover the ankle, and completes the wiping over the socks, then the ruling applies to both of them together.
If he takes off the shoe only, or takes it off with the socks, that does not render his wudu invalid and it is permissible for him to pray, but it is not permissible for him to wipe over them again after that, unless he has done wudu in full and washed his feet.
Almighty Allah knows best.