The permissibility of performing wuḍūʾ in one mosque and praying in another

Answer—and Allah is the One who grants success: As it appears to me, there is no objection to that, because wuḍūʾ is an independent act of worship, as evidenced by His saying, Exalted is He: “Indeed, Allah loves those who repent and loves those who purify themselves.” [al-Baqarah222], and his saying (may Allah bless him and his family and grant them peace): “Wuḍūʾ is half of faith.”
This—and perhaps what prompted the questioner to ask this question is that the one who set the water in the mosque intended it for those praying in that mosque; thus, one who performs wuḍūʾ and then prays in another mosque has used the water for other than what it was set for, and that is not permissible.
We say regarding that: wuḍūʾ is an independent act of worship, as we mentioned. Therefore, one who uses that water set in a particular mosque for wuḍūʾ and removal of major ritual impurity is not overstepping what the donor designated it for; indeed, the donor set it for that very purpose.
That which is not permissible is to use the water for purposes other than that for which it was provided—such as diverting it to houses or watering fields. As for when the water is so scarce that, as a rule, it suffices only for the worshipers in that mosque, then it is not proper for others to come merely for ghusl or wuḍūʾ. However, if someone comes for wuḍūʾ and prayer, he has a right to that water, because the one who placed the water in that mosque intended thereby to keep that mosque alive.
Source : Min Thimār al-ʿIlm wa al-Ḥikmah vol.1