Yes. They are used to move quickly from east to west, hit and go with booty.
Al mundhir of the nasrides used to ride the roman area to check the roman resistance.
In the 7th century, they have defeated the persians.
And in the same time (the 30's) went west with one clear first objective which is linked to their precedent (5,6 th. c) expeditions.
They will continue to attak the west as they did already independently of persians.
The eastern persians and western roman arabs had to be addeded in the peace treaty to stop fighting each other without their masters permission.
Romans are retreating, the western arabs were disbanded by mauricious. They take over the political power after defeating the romans at yarmuk (or at least pushing a retreating army away)
It's more simple than that : Eastern Arabs went west: what was their first objective? (reflect, it is very simple when you link it to what they did before)
They head north and then pass by jerusalem in 637 and build the first house following quranic verses.
Then they defeat their eastern foes in 661.
Those who " head north and then pass by jerusalem in 637 and build the first house following quranic verses." and installing in Jerusalem-Damascus come from the East ; they are Iraqi. As such they do not want to lose Iraq ; that is why from Damascus where they are installed, they continue to look to Iraq because it is their homeland.
The question is how do we understand how they came to find the quranic texts.
I have a response to this (important ) question
but I will not giving it here.
The western arabs were led by the jafnids who were miaphisites.
Now there is another group leading them with quranic texts.
Both eastern and western arabs looks like being in possesion if quranic texts.
I do not think so. For me, only the Easterners had (some) quranic texts.
What do you think of the book of jeremiah as inspiration and source for quranic muhammad
It is interesting. But accessory as all Biblical prophets had more or less the same adventures. I consider that the pattern-source on which 'Muhammad' was built is another (Biblical) figure of the Quran.