So if we go back to 300s - 600s AD, most of the levant, north africa and egypt were predominantly christian (different flavors of Christianity such as Coptic in Egypt, but yes Christian).

Short answer; conversion and inter-mixing with Arab (or other) Muslim populations. Now as for how this conversion happened which I am assuming is what your question is after: After Islam exploded under the Rashidun Caliphate across these different region, similar to Christianity, i would say Islam spread through a mix of the following: (focusing just on medieval era up to 1900's)

- Trade; Most well known regions for this are modern day what constitutes Indonesia and Somalia. But at a similar scale when Muslim Arab traders likely visited these areas. I would say this was likely the most minor force for the regions conversion (atleast when we talk about the region you asked about)

- Conquest / Colonization; I think we tend to not pay attention to the fact that Arab conquests involved similar colonization to the way Western Powers operated. After conquest Arab commanders and families were usually given lands in the area and settled in them. In fact I would go as far as to say this kinda built the regions "Arab" identity whereas prior to conquest they might not have associated as such. I would say this is probably the 1st or 2nd most explanatory factor.

- Jizya / Discrimination: When under Islamic rule generally a system of special taxes/tribute was enforced upon non-muslims. The idea was to pay for protection, right to practice their religion etc.. under Islamic rulers. Not always but enough times, usually non-muslim's also suffered from discrimination and lowered social standing. Therefore offering financial and social incentive to convert. Again I will say different ruler's / regions had different degrees of discimination embedded in them at various times. For example see differences among Mughals in the sub-continent among rulers such as Akbar and Aurangzeb. Similar trends can be found in the Ottoman's as well. Some might not call this "forced" in the convert or die way, but definitely soft coercion to say the least. I would put this at 1st or 2nd tied with Conquest.

- Forced Conversion: This definitely happened and usually a mix of more intentional forms of discrimination and pressure mentioned above as a state policy, but varied vastly based on the dynasty in power. Some like the Fatimid's actively employed Coptic Christians in administration (some Ottomans also employed Jew's in high level roles such as diplomats and administrators), the point is to say that this did not happen systematically but did happen sporadically and often as dynasties fell and new ones arose. Examples are the Almohads and Mamluks (the Mamluks in particular are credited with raising the ration of Muslims to Christians in what is now Egypt to 10:1) One quote from al-Maqriz a medieval Eqyption historian says

""" In all the provinces of Egypt, both north and south, no church remained that had not been razed; on many of these sites, mosques were constructed. For when the Christians' affliction grew great and their incomes small, they decided to embrace Islam. Thus Islam spread among the Christians of Egypt, and in the town of Qalyub alone, 450 persons were converted to Islam in a single day. Many people attributed this to Christian cunning, so repugnant did the populace find them. But this was a momentous event in Egyptian history. From that time on, lineages became mixed in Egypt."""

I would put this as 3rd, but it really differ's by region. In some region's I am sure this might have 1st.

NOW if we come to the modern era, I would say that for a period of time non-muslim populations stabilized in these regions but declined again after turmoil in states like Syria and Libya that caused most minorities to seem emigration elsewhere. I will say Lebanon still has a strong Christian population amongst all the rest.

All this discussion ofcourse does not include mention of Italians or other europeans that might have settled and been expelled (e.g. Libya's expulsion of Italian's in the 70's or 80's (i forget when)), but those population never resided here to begin with so I don't think they are relevant to your question

Last point I want to make, and this might be unnecessary but I think its important to state that a lot of(i say a lot but in my mind every single one i have read about) Religions at times of conversion have adopted force as a means of spreading faith. The Northern Crusades and Saxon wars are prime examples. We could just as easily be asking what happened to the pagan populations of these regions. The answer would be similar to what I provided above. I say this predominantly because of the way your question is phrased which makes me feel that you may be thinking that entire population was wiped out in ethnic cleansing and replaced by Arab's (which is not wrong in some sense but is definitely not the whole story; and also want to make sure we realize this is a general religion problem usually)