Praise be to Allah.
If the person who is continually breaking his wudu– such as one who is suffering urinary incontinence – renews his wudu before the time for prayer is over, he does not have to wash his private parts when renewing it, because he has wudu according to the ruling.
But if the time for one prayer ends, then he has to wash his private parts and do wudu again, then pray as much as he wants of obligatory and naafil prayers (during the time for the next prayer), because the Prophet (peace and blessings of Allah be upon him) said to Faatimah bint Abi Hubaysh [who was suffering from prolonged vaginal bleeding], “Wash the blood from yourself then pray, then do wudu for each prayer until that time [i.e., the usual time of menstruation] comes.” Narrated by al-Bukhaari, 226; Muslim, 333.
If we assume that he has done wudu, then nothing comes out of him until the time for the next prayer begins, then he does not have to do wudu again, and his first wudu is still valid.
The words of the fuqaha’ – that he should do wudu at the time of each prayer – apply in cases where something has come out of him.
Al-Bahooti said in al-Rawd al-Murabba’ (p. 57): “The woman who is suffering from istihaadah [non-menstrual vaginal bleeding], and others like her such as those who are suffering from urinary incontinence, continual emission of madhiy (prostatic fluid) or wind, a wound that bleeds continuously, or a continual nosebleed, should do wudoo when the time for each prayer begins if something has come out, and they should pray so long as the time for prayer lasts, both obligatory and naafil prayers. If nothing has come out then they do not need to do wudu.
And Allah knows best.