Cushitic and Semitic are two linguistic branches in the Afro-Asiatic language family. From a historical perspective, people of various disciplines and backgrounds had the tendency to associate tribes or speech communities with their branch, as that informed substantially on culture and relational context, and there is the overall strong (not absolute), the correlation between genetic affinities between language and genetics. Semitic-speaking Horn of Africans is a good case study and convenient for the thread, so let's take and explain them in simple terms.
Besides a few irregular, highly admixed peoples that belong to Cushitic (excluding Omotics completely to reduce the complexity of explanation and reduce uncertainty level), the overwhelming majority of the DNA Cushitic speakers carry came from north of the Horn of Africa. Early Cushitic tribes around 7000-5000 before present migrated southeast, and additionally absorbed relatively substantial minor indigenous hunter-gatherer DNA. Now we have a genetic structure that we today consider the essential component structure that makes up a "pure" East Africa Afro-Asiatic people.
As time went by, particularly around 2800 before the present, South Arabian tradesmen migrated to modern Eritrea and northern Ethiopia and settled. These people mixed with local Cushitic populations that were Agaw-like in speech and influenced linguistically as well for them to speak the first Ethio-Semitic language. From a temporal perspective, as time went by, new languages and dialects formed out of this initial language, and the South Arabian-derived ancestry frequency got stabilized between the Ethio-Semitics to exist in the ~25% range (it can fluctuate 5% above and under for good measure).
The question of relatedness in the Horn of Africa is driven by geographic association. This subject matter is can be complex; there are always exceptions to the rule, so I will keep it simple and generalized. Oromo people are heterogeneous in that they can exist in three clusters and anything in-between. We have Oromo that got substantial Omotic admixture (Oromos in general, no matter what cluster, still got higher hunter-gatherer admixture levels compared to the average Semitic-speaker, Cushites from Eritrean, northeast Ethiopian Cushitic-speakers and Somalis), that are on autosomal (genetic makeup) basis, no different from Omotics (Afro-Asiatic people with substantial endemic hunter-gatherer ancestry). We have Oromos who were acculturated—genetic profile-wise no different than an Amhara—yet is Oromo on an identity basis. And then you have Oromos that show a good amount of affinity to Somalis, simply put.
Oromos then will show an unequal relation to Somalis depending on what Oromo you choose to compare with. But the overwhelming majority of Oromos, even the acculturated ones that show Semitic-like profile will have a closer affinity to Somalis than a Tigray Habashi or Eritrean are to Somalis, genetic distance-wise. Now, although, on a very smaller scale, the Afar shows their own peculiarity. Linguistically they are Cushitic, on aspect about DNA they, in Ethiopia at least, show no difference from Ethio-Semitic peoples - having elevated Arabian element. But I suspect they, in Djibouti, for example, will have an excessive convergent affinity with Somalis. In this case, it is not too far off to guess that some Afar will be closer to Somalis than some Oromos, even excluding the Omotic-profile ones. One thing with Ethiopian populations is that genetic closeness is dictated more by proximity than linguistic barriers. That is the trend I see on a general basis.
On the matter of Beja. These folks, you sort of have to analyze by themselves because they got a different historical process situating from Egypt, along the Red Sea of the Eastern Desert of Sudan, all the way to Eritrea. So Beja's got influenced by their own set of Cushitic original DNA, gene flow between fellow Sudanese, and northernmost Habashi people of Eritrea. To what level these people got Semitic ancestry, namely of the Peninsular heritage, is dependent on other factors and it varies. But they, in a lot of ways, got high Arabian ancestry. Beni Amer, as an example, carries higher Arab than Habashi people.
Someone should move this thread to Culture & History.