I think you're focusing on West of today's Saudi Arabia, rather than East.
In relation to Mohamad & his followers, take a look at the section entitled "Raids on Caravans - Caravans & Trade as a Source of Wealth in this page
You are quite correct that we tend to pay more attention to the Muslim conquests west rather than east. Perfectly logical, since the western conquests moved into areas with which we have a historical and civilizational connection. They were Christian and Western in character (at least sort of) up to the Muslim conquest. Persia and Central Asia were, and are, the Other.
I have no doubt whatsoever that Arabs raided into Syria and Iraq all down through history whenever the got the chance. Given the inherent disparity between the populations and military potential of the settled areas versus the deserts, large raids only took place when the settled areas were in great disarray. Which probably in all history were never greater than when the Byzantines and Persians had battered each other into exhaustion after centuries of intermittent war and 30 years of continuous war. Each contender had invaded and devastated the other’s heartland. The effects were probably right up there with those of the 30 Years War on Germany, or possibly even worse.
The problem is that the mindset needed for effective conquest is very different from that of raiders, who seldom set up governments that last very long. As Napoleon said, you can do anything with bayonets except sit on them. Conquest requires structure, and smash and grab bandits aren’t big on structure.