Though not specifically addressed in the book I'm reading (The Silk Roads), the spread of various religions throughout the civilized world is fairly explainable by the routes it took.
In pre-industrial days, there was much in the world that could not be explained. Crop failures, droughts, hurricanes, earthquakes, pestilences, eclipses, illness, death.
Those who claimed to be able to explain the unexplainable had instant higher status. And if they claimed to be close to the controller of all these unexplainables—if they claimed to have sway with the unseen master—they would be sons of gods and more than mere men.
Every time a new religion rolled around, if it claimed to be more accurate in controlling crop failures or child mortality than the old religion, which had proven itself largely ineffective, then the new religion would become everyone's new favorite.