What is meant by the Prophetic envoy? Is it only about delivering the message, or does it also involve staying in the country?

In Arabic, a "safeer" (envoy/ambassador) is one who reconciles people. As for the Prophetic envoy, he is someone whom the Prophet, peace and blessings be upon him, sent as a messenger, a teacher, or a caller to an affair of the religion. The envoy might be sent to reside where he was sent, like Mus'ab ibn Umayr who stayed in Madinah to teach its people Islam, or he might be sent carrying a letter and then return, as happened with the Prophet's envoys to kings and princes.

Envoys were characterized by several qualities, among them: a. A precise and comprehensive understanding of Islam. b. Extensive experience with the regions they went to and the people they spoke with. c. Good character, patience, and consideration for the rights of those addressed. d. The ability to face sudden situations and act appropriately. e. Good understanding, speech, and presentation. f. Sincerity in belief and clarity in presenting their call. g. Personal courage stemming from their trust in Allah.