I understand that missionaries were some of the first people to arrive in different places during the age of exploration. Nations were on the search for "Glory, Gold, and God", meaning they were seeking to expand their lands, earn a lot of money doing it, and convert the locals. Sometimes the natives would see the conversion to Christianity as an opportunity for them to become wealthy or more powerful locally and this would lead to at least a portion of them converting fairly quickly. Not to sure about Islam though, I know that conquered places such as Andalusia and Egypt were forced to convert to Islam after they were conquered, and this was the same thing in India which was almost entirely conquered by the Muslims on several occasions.