In a weird sense, Saudi could be considered a constitutional monarch where the Qur'an / Sharia are the constitutions of the country. As applied, those don't really protect rights as most constitutions do.
In order for Saudi to become a traditional constitutional monarchy, it would probably have to flow like most other such -- from a revolution/uprising that forced concessions (think Magna Carta) but didn't fully overthrow things.
Other conservative monarchies COULD become constitutional monarchies the same way. However, there are some things that lower the probability:
1) In many cases, such an uprising would completely overthrow the monarchy and would go to a democracy of some sort (Arab Spring -- Egypt, Libya, etc)
2) In some cases, uprisings in that area might essentially become radical Islamic dictatorships (like ISIS) that could eventually become monarchies where the Qur'an / Sharia are essentially considered the constitutions of the country
As it says in the tin. What would it take for Saudi Arabia to become a constitutional monarchy? Is it ASB or not?
Also, about the other Arab countries with conservative monarchies, how would they democratize?