Other groups have been accreted to the "True Arabs", of course, and in the end the Arab language, and a lot of Arab culture, was spread all over the place.
Then there are the Jews (as they became known after the Kingdom of Israel was divided and the Northern tribes "taken captive").
They founded colonies in various places ~ sometimes after having been conquered by others, or voluntarily ~ usually as trading ventures.
Since many people besides "True Arabs" speak Arabic, it's simply not the case that all speakers of Arabic are "True Arabs". In fact, Classical Arabic, which became a literary language of some note, is probably more Aramaic (a Jewish language) than it is Arabic in its origins. A recent thread here pointed to a modern analysis made of the Koran that suggests that it was originally a Syriac-Aramaic document, and is more properly read in Aramaic than in Arabic ~ you get quite different interpretations that way, too!
But, rolling things backward, in Abrahams time his people were undoubtedly Sumerian and didn't even speak a Semetic language. After a sojourn in the Levant, and then in Egypt, where the tribes acquired brides (Joseph and his brothers did that FUR SHUR), they became a semitic speaking group.
At some point it has always been argued they acquired more than their fair share of Armenoid characteristics from the Armenians, as well as more than their fair share of redheads with blue eyes from the Kurds (also called the Medes in the Bible).
The history of most of the present day population of Palestine is pretty much the same, and it's really ridiculous to argue that they just recently rode in from the desert. The DNA shows they didn't!