Is it permissible to offer the 'Asr prayer before the missed Dhuhr prayer if there is only enough time left in the 'Asr prayer's designated period to perform one prayer, in order to catch the 'Asr prayer before its time expires, or is it obligatory to observe the proper sequence when making up missed prayers?

Arranging prayers in their proper order is obligatory according to most scholars. However, if one fears missing the time for the current prayer, then the order is dropped, and the current prayer is given precedence. This is the view of the majority, including Sa'id ibn al-Musayyab, Al-Hasan (definition), Al-Awza'i, Al-Thawri, Ishaq, and the People of Opinion. Al-Muwaffaq said in Al-Mughni: "If one fears missing the time [of the current prayer] before making up the missed prayer and repeating the one he is in, then the order is dropped at that time, and he completes his prayer and then makes up the missed prayer." The evidence for this is that Allah commanded the performance of the current prayer within its time. Giving precedence to the current prayer keeps it as a performance, whereas if the missed prayer were given precedence, both prayers would become make-ups. So, whoever fears missing the time for Asr and has not prayed Dhuhr, should begin with Asr and then pray Dhuhr, and the order is dropped for him.