Assalamu alikum wa rahmatou Allah wabarakatoho.
Dear brothers, my question is concerning a tradition we have here where I live. People usually meeting each other greet using SALAM than shake hands. While shaking hands they say : "Allahoma sali wa salim ala sayidina Mohammed". Is doing so permitted ? Or it's some bat tradition that we must refrain from doing?
May Allah bless you all, and guide us all to his righteous path...
Peace be on you.

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.

The virtue of invoking blessings upon the Prophet, sallallaahu ‘alayhi wa sallam, is great, and its reward is enormous, so one should repeat it as much as possible.

However, invoking blessings upon the Prophet, sallallaahu ‘alayhi wa sallam, in the mentioned situation in particular or in any other situation other than the reported regular acts of Sunnah, while believing that it is an act of Sunnah or that it holds a special virtue, is impermissible because it is an innovation which belongs to what is known as additional innovations. For more benefit, please refer to Fatwa 116190.