Take a look at this animation of how the 5 major religions of the world spread.

Both Christianity and Islam spread in large part because one of their doctrines is to spread the religion.

Take Judiasm for example. Many Jews believe that the jewish people (ethnicity not religion) have a covenant with god. If a Native American or Japanese person can not become a jew (they are not of jewish ethnicity), why would you spread it there? This was a main reason Judiasm did not spread very far.

Proselytizing is much more prominent in Christianity and Islam than most other religions too.

The reason for christianity specifically is because of European Colonialism. With Christianity and Islam, generally once they converted a population, the population rarely changed sides. Christians got the Americas first and Muslims got Northern Africa first. No Hindus or Buddhists go there to try to get them to change, so why would they?

Are you looking for generalized answers covering the last 2,000 years (which I wouldn't feel qualified to fully answer) or would you also prefer just discussing reasons why Christianity went from being an unknown Jewish Sect to the dominant religion in just over 300 years?

Your topic is really too huge for any real analysis. You are essentially asking someone to meta-theorise a 'unique factor' that lead to religious change across various cultures, languages, people and time periods for around 1900 years.