What is the ruling on traveling to visit the grave of the Prophet (peace be upon him) and other graves of saints and righteous people?

It is not permissible to travel with the intention of visiting the grave of the Prophet (peace and blessings be upon him) or any other grave, due to the Prophet's (peace and blessings be upon him) saying: "Journeys are not to be undertaken except to three mosques: Al-Masjid Al-Haram (definition) (the Sacred Mosque in Makkah), this mosque of mine (the Prophet's Mosque in Madinah), and Al-Masjid Al-Aqsa (in Jerusalem)." The prescribed action for someone who wishes to visit the noble grave and is far from Madinah is to intend by their journey to visit the Prophet's Mosque, whereby the visit to the grave then becomes secondary. However, if one is near and does not need to undertake a journey, there is no harm in that. Visiting his grave and the graves of his two companions without undertaking a journey is a Sunnah (definition) and an act of worship, and likewise, visiting the graves of the martyrs, the people of Al-Baqi', and all other Muslims is a Sunnah and an act of worship, but without undertaking a journey. The Prophet (peace and blessings be upon him) used to teach his companions, when they visited graves, to say: "Peace be upon you, O inhabitants of these dwellings, believers and Muslims. And indeed, Allah willing, we shall join you. We ask Allah for well-being for us and for you."