Did the Prophet, peace and blessings be upon him, weep out of longing to see us, as the widespread saying indicates: "Send prayers upon him who wept out of longing to see us"?

The phrase "the Prophet's weeping out of longing for the Muslims who would come after him" is not accurate. Rather, what is established is that he wished to see his brethren among the believers who had not yet arrived, as mentioned in the hadith narrated by Abu Hurairah and recorded by Muslim.