Answered by Shaykh Abdul-Karim Yahya, SunniPath Academy Teacher
Is it permissible, and even recommended, to send blessings on the Prophet (Allah bless and grant him peace) during the bow (ruku) and in prostration (sujud), such as saying ‘Allahumma salli ‘ala Muhammad wa ‘ala aali Muhammad’ or ‘Allahumma salli ‘ala sayyidina Muhammad wa ‘ala aalihit tayibeen at tahireen wa azwaajihi ummahat al mu’mineen wa khulafaa ar rashideen al mahdiyin wa sahabatihil gurril mayameen’ for a long version? If it is permissible, can it be done in the obligatory and voluntary prayers, because I know in the Hanafi School it is not allowed in the obligatory prayers? Also, what other invocations can be made while bowing and prostrating?
In The name of Allah, Most Merciful and Compassionate
During the prayer it is not recommended to send blessings (salat) on the Prophet (may Allah’s purest blessings and peace be upon him) in other than its prescribed place (i.e. after tashahhud or qunut) because it is a spoken integral of the prayer.
Deliberately moving a spoken integral to other than its place does not invalidate the prayer according to the soundest position, but it is sunna to prostrate for forgetfulness if one does so.
Invocations for bowing and prostrating are mentioned in The Reliance of The Traveler. It is generally sunna to beseech Allah in prostration due to the tradition narrated by Muslim: “The nearest a slave is to his lord is when he is prostrate so supplicate earnestly.” i.e. in prostration and invocations transmitted from the Prophet are best. Allah bless him and his folk and grant them all peace. And Allah knows best and He alone gives success (tawfiq).
 See: Yahya Bin Sharaf al-Nawawi, Minhaj al-Talibin Wa ‘Umdah al-Muftin(Eygpt: Sharikah Maktabah wa Matba’ah Mustafa al-Babi al-Halabi wa Awladuhu, 1338 A.H. [1920 C.E.]), p.14.
 See: Ibn Hajar al-Haytami, Hawashi al-Sharwani Wa Ibn Qasim al-Abbadi ‘Ala Tuhfah al-Muhtaj Bisharh al-Minhaj (Beruit, Dar Ihya’ al-Turath al-‘Arabi), 2:103.