All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
Let us say, first of all, that it is proven that the Prophet, sallallaahu ‘alayhi wa sallam, forbade that one prayer should be joined with another, as shown in the Hadeeth narrated on the authority of Mu‘aawiyah in which he said: “The Messenger of Allaah, sallallaahu ‘alayhi wa sallam, commanded us to do so, i.e. not to join two prayers before talking (between them) or going out.” [Muslim, Abu Daawood and others] So, if you finish ‘Asr or any other prayer and intend to perform another prayer, you should not pray directly after reciting the Tasleem of the previous prayer before you talk or at least move from your place.
As for performing the prayers referred to in the question after ‘Asr, then offering voluntary prayer after ‘Asr is forbidden, in principle, due to the statement of the Prophet, sallallaahu ‘alayhi wa sallam, “There should be no prayer performed after ‘Asr until the sun sets.” [Al-Bukhari and Muslim]
However, scholars have different opinions over offering the prayer for which there is a reason during the time in which voluntary prayer is forbidden. The opinion that we adopt here in Islamweb is that it is permissible to perform it at that time (of forbiddance). By acting upon this opinion, the different texts could be reconciled, and this is the opinion of Imaam Ash-Shaafi’i and it is also the choice of Shaykh Al-Islam Ibn Taymiyyah . Some jurists regard the prayer of repentance among those prayers for which there is a reason, which may be missed unless it is performed at its due time, and it could be performed at the time of forbiddance.
In this connection, Shaykh Al-Islam Ibn Taymiyyah said while stating that the voluntary prayer for which there is a reason could be performed at the time of forbiddance: “All prayers for which there is a reason would be missed if they are postponed until the time of forbiddance is over, like the prostration of recital (Sujood At-Tilaawah), the prayer of greeting the mosque, the eclipse prayer, and the prayer that is performed just after Tahaarah (ablution), as in the Hadeeth narrated on the authority of Bilaal and the Istikhaarah prayer, if the object for which you seek Allaah's guidance would be missed by postponing the prayer. The same applies to the repentance prayer: if you sin, repentance becomes due immediately. It is recommended, in this case, to perform a two-Rak‘ah prayer and then repent.” Hence, there is no harm in performing the prayer of repentance, when its reason exists, after ‘Asr.
Also, there is also no harm if the one who misses the pre- Sunnah (supererogatory prayer) of ‘Asr prays it after ‘Asr, because it is proven that the Messenger of Allaah, sallallaahu ‘alayhi wa sallam, made up the pre-Sunnah prayer of Thuhr after ‘Asr.
As for the two-Rak‘ah prayer mentioned in the Hadeeth on the authority of ‘Aa’ishah that the Prophet, sallallaahu ‘alayhi wa sallam, performed it regularly after ‘Asr, it is exclusive to him, sallallaahu ‘alayhi wa sallam, in the sight of many scholars, because ‘Aa’ishah was asked about the two-Rak‘ah prayer the Messenger of Allaah, sallallaahu ‘alayhi wa sallam, used to perform after ‘Asr, and she said: “He used to perform it before ‘Asr prayer. Then, he was busy or forgot it, thereupon he prayed it after ‘Asr prayer. Since then, he continued to perform it regularly; and it was his habit that whenever he performed a prayer, he would continue to perform it regularly.” Thus that he performed it regularly is exclusive to him, sallallaahu ‘alayhi wa sallam. So, it is not valid, under Sharee‘ah, to perform a supererogatory two-Rak‘ah prayer after ‘Asr if there is no reason for it.
Allaah Knows best.