I understand that we should strictly follow the Sunnah of the Prophet (sws). It is stated that he used to visit graves. I would be grateful if you could enlighten me as to what he used to do at the graveyards?

One of the prayers that the Prophet (sws) is reported to have made while visiting a graveyard was in these words:

Peace be on you, O! the residents of graves. May God forgive you. You have preceded us and we are following your footsteps. (Tirmadhi, No: 1053)

Since graves of relatives are an effective reminder of death, the Prophet (sws) encouraged believers to visit them frequently. Of course, prayers were also rendered for the forgiveness of the dead and for raising their status in the Afterlife.

What I may have emphasized is that no prayer of the Prophet (sws) for the dead contains a mention that indicates that he asked the Almighty to transfer any of his own deeds in favour of any deceased person. That is something completely against the principles of Islam. And Allah knows best.

Courtesy: http://www.islamicissues.info/qa_question.php?qid=203