It depends what time period, but also note that archaeological and historical research into Aksumite to post-Aksumite Ethiopia has had a bias toward looking at Ethiopia's relations with the Classical World and the West. There has been a little work on Aksumite period contacts to the north along the Nile, but not much else across the region beyond networks through the Red Sea to the Middle East and Mediterranean. A Malian colleague and I used to joke in grad school we would run a collaborative project to study east-west trade between the great West African empires he studied and the Ethiopian state (Abyssinia was always more of an ethnic term and at least by the 20th century, Haile Selassie and others argued Europeans should please stop using this term).

The short answer to your question seems to be no, Ethiopia never had too much direct contact or express much interest in populations too far beyond its frontiers with the African interior (south to west). This may in part be simply omissions from internally recorded history, and/or it may have some to do with Ethiopian interests and power. Proselytizing, on the other hand, was a slightly different matter and I’ll treat that separately. It also bears keeping in mind that Ethiopia as it exists today is a product mostly of the 19th century. The Aksumite and Solomonic states/empires historically only extended about as far south as the Awash Valley that bisects the modern country, so modern areas like Uganda were divided from Ethiopia by historical areas like Gojjam and the southern Ethiopian Highlands that often formed the state's frontiers or peripheral vassal states.

During the D’MT and Aksumite periods (c. 800 BCE – 700 CE) the region was heavily oriented toward the complex economies of the Red Sea, Middle East and Mediterranean worlds. What if any interest they had toward the interior probably only took the form of down-the-line trade rather than sending envoys or caravans too far beyond their own borders. There is an account, the source of which I am blanking on, that describes “silent trade” where Ethiopians would arrive at the frontiers with their neighbors and go through a customary process of bartering for goods from one another’s lands.

During the immediate Post-Aksumite period, there is a progressive southward migration of the political center of the Ethiopian state and a gradual expansion of the states and territories of Ethiopia. This expansion is soon and greatly checked, however, by the mass migration north of pastoralists (called Beja at the time, but Oromo is more proper today) through the Southern Highlands, across the Awash, and into the northern highland territories of the Solomonic Dynasty. The Oromo migration would continue to be a huge source of tension in Ethiopia and politically gradually eroded the power of the imperial system such that even if they wanted to make direct contact with populations beyond their immediate neighbors, they would have met with great difficulty.

Ethiopia was also a lot less wealthy by this period because the spread of Islamic states and the decline of the Eastern Roman empire/Byzantium shifted global trade networks east while Ethiopia was largely cut off from its earlier ports. Goods that once made Ethiopia wealthy like ivory and gold, were becoming much easier to acquire from the burgeoning Swahili coastal cities to the south, which were adopting Islam and thus ingratiating themselves into Indian Ocean trade networks. This led Ethiopia to increasingly rely on Muslim caravans for its external trade moreso than its own internal efforts. Pedro Paez in his History of Ethiopia, 1622, describes the almost extortionary relationship by which the Ethiopian emperors compelled Muslim-run caravans to do trade on his behalf. All the while in this period, tensions are occasionally flaring up with neighboring Muslim states like Ifat and Adal, primarily to the southwest, rather than the north, further distracting any potential Ethiopian interest in making contacts with the African interior. What external contacts Ethiopians did seem to be concerned with remained oriented toward the Mediterranean world. As a Christian nation they were beholden to the Coptic patriarchy in Alexandria/Cairo, Egypt, so frequently maintained contacts in that direction; likewise, Ethiopians maintained a vested interest in Jerusalem and the Holy Land.

Basically, it seems the Ethiopians had a general disinterest in what was going on too far beyond their southern frontiers. Their interests were primarily toward the north and what contact they did have toward the south were probably through intermediaries like externally operated caravans or down-the-line contacts through their neighbors. There were also pressures like the northward Oromo migration and frequent conflict with neighboring sultanates to make southward interests/expansion a hassle.

Likewise, the imperial state notably seemed disinterested in spreading the word of Christ unlike other early medieval European Christian states. I’m not sure anyone really has an explanation why, but it simply does not seem to have been high on their agenda. The emperors even seemed rather reluctant to travel beyond the core of their empire as their recorded itineraries show unless there was a compelling reason like conflict to do so. This does not mean proselytization did not occur, however! Mobile priests, monks, and deacons were a notable feature of Christian Ethiopia. Church and monastic communities frequently grew so large they spawned daughter communities that would produce ever-expanding networks of Christian communities with some sense of fealty toward their mother communities. Deacons (dabtara) were also sort of the mendicant medicine men of their day and regularly were so numerous they could not be housed at a single clerical community indefinitely, so they traveled. Steven Kaplan in his book, The Monastic Holy Man and the Christianization of Early Solomonic Ethiopia (1982) argues at length that monks and deacons were at the forefront of the expansion of Christianity south of the Ethiopian state. They would move beyond Ethiopia’s immediate frontiers seeking room to establish new clerical communities and, incidentally, to preach to the natives while there. As these nascent clerical communities gradually Christianized the area, Ethiopian emperors would often come behind them and try to incorporate these newly Christianized frontier communities into their empire.