Excuse me, we don't worship shrines, we don't worship even the prophet (SA), we only pray to Allah, other than that keep your beliefs to yourselves in your countries, away from Muslims. As far as I know, prophet's grave is well taken care of under the custodians of the two holy mosques.
Since you brought in desecration of holy graves I will post this to readers to decide who realy are known of doing that, and BTW, this is written by a Westerner analyst named Martin Kramer, criticizing both sides, however:
Khomeini’s Messengers in Mecca
From Ottomans to Saudis
The Sunni corpus of libel is perhaps more readily documented, if only because it sometimes led to violent acts against Shi‘ite pilgrims.
At the root of the Sunni lore is the belief that Shi‘ites feel themselves compelled to pollute the holy premises. Much evidence for Sunni belief in this libel exists both in Islamic textual sources and in European travel literature. This pollution was said to take a particularly repelling form: Burckhardt and Burton, the great nineteenth-century explorers of Arabia, both heard about attacks on Shi‘ite pilgrims,
prompted by the suspicion that they had polluted the Great Mosque in Mecca with excrement. According to Burton, “their ill-fame has spread far; at Alexandria they were described to me as a people who defile the Ka‘bah.”4
The Safavids reacted by trying to discourage the pilgrimage to Mecca and emphasizing the importance of Shi‘ite shrines in their own domains.3
All the more striking, then, was a serious recurrence of the Sunni libel of Shi‘ite defilement. In 1943, a Saudi religious judge
ordered an Iranian pilgrim beheaded for allegedly defiling the Great Mosque with excrement supposedly carried into the mosque in his pilgrim’s garment. Ibn Sa‘ud remarked to some Americans that “this was the kind of offense which might be expected of Iranian.” The verdict in local coffee houses held that “the Iranians always act that way.”17
Khomeini’s Messengers in Mecca | Martin Kramer on the Middle East