I'm really interested in the period before European arrival, but as I imagine that's a bit more difficult to source for English-language historians, I'd also be interested in the early period of European arrival in the area (before they were actually conquered by Portugal, Britain, etc).
In response to a comment now deleted, I realise that "the interior" is not very clear. I mean the tribal groups inland, in what is now Kenya and Tanzania. The ones that might have some contact with the Swahili city-states. Not all the way into the central African jungle.
At the most basic level, relations between Swahili city-states and the hinterland were a tension between peaceful trade and conflict provoked by slave raiding.
Ibn Battuta presents a dismal picture of coast-hinterland relations in his account of a visit to Kilwa Kisiwani Masoko* in the 1330s.
Kulwa is a very fine and substantially built town, and all its buildings are of wood. Its inhabitants are constantly engaged in military expeditions, for their country is contiguous to the heathen Zanj.^1
As well as
He [the sultan of Kilwa] used to engage frequently in expeditions to the land of the Zinj people [villagers of the interior], raiding them and taking booty [slaves and other wealth]^2
However, archaeological surveys since the 1970s have presented an entirely different picture. In Kenya's Tsavo National Park, P.F. Thorbahn found evidence of sustained coast-Tsavo contact from AD 1200, and he hypothesized that Tsavo hunters had become specialists in hunting elephants to supply ivory to the coast. Chapurukha Kusimba also points to the presence of imported pottery from Persia, China and India in the homes of coastal peoples. He speculates that coastal elites (swahili) may have used coercion to control trade with Tsavo agripastoralists, with greater ability to control Tsavo trade as social complexity increased in the coastal towns after 1200.^3
Kusimba went on to propose a "cultural mosaic" present in the Tsavo region, with similar developments pointed to along the Rufiji river and in south-central Tanzania^4. His Tsavo Cultural Mosaic proposes that various groups of merchants (Swahili, Mijikenda), hunters (Boni, Dahalo, Waata), pastorialists (Oromo, Wambisha, Somali) and agropastoralists (Wakamba, Taita) all traded among each other, and pursued food-getting and economic strategies that complemented each other, within their distinct ecological region.^4
He points to archaeological surveys done by P. Msemwa along the Rufiji watershed that suggest a region-wide interaction sphere between 400-600 AD^5. Given that there is evidence of trade in the Rufiji region going back to the Roman era^6, this raises the possibility that Swahili traders of the period from 600-1500 AD were "plugging-in" to coastal trade networks that pre-date a distinct Swahili identity.
Archaeological investigation into the coastal-hinterland is very much an ongoing concern. Further surveys may uncover evidence that extends the reach of coastal trade further inland. However, at present it seems that Swahili traders did not reach inland regions like Uganda before the mid-19th century. The first British christian missionary accounts in the Buganda kingdom mention a nascent Muslim community in the kingdom, Islam having been introduced by Swahili traders only a generation before.^7 However, the 19th century saw the massive expansion of Swahili trade networks. Tippu Tip is known to have traveled from Zanzibar to the eastern half of what is now the Democratic Republic of Congo to use as a hunting ground for slaves.
1 Ibn Battuta: Travels in Asia and Africa 1325-1354 "kulwa on the African mainland"
2 Travels of Ibn Battuta. This site uses the G.H. Gibb translation of Ibn Battuta, which is slightly different from the Fordham version above. I quote the 2 different versions because the Fordham version simply says "the sultan gives away one fifth of his wealth" but leaves out the context of it being booty and slaves.
3 "Hinterlands and cities: Archaeological investigations of economy and trade in Tsavo, southeastern Kenya" by Chapurukha Kusimba and Sibel Kusimba in Nyame Akume No 54, pp 18.
4 "the Development and Collapse of Precolonial Ethnic Mosaics in Tsavo, Kenya" by Chapurukha Kusimba, Sibel Kusimba and David Wright, in Journal of African Archaeology Vol 3, no 2, 2005. pp 245.
5 Ibid, pp 245-246.
6 "Roman Beads from the Rufiji Delta, Tanzania: First Incontrovertible Archaeological Link with the Periplus" by Felix Chami, in Current Anthropology vol 40, no 2. march 1999, pp 237-242
7 White Fathers, Islam and Kiswahili in Nineteenth-century Uganda PDF warning. Please note that "At the time of Stanley’s visit in 1975" on page 199 should read "At the time of Stanley's visit in 1875"
Edit: Kilwa Kisiwani is the name of the island settlement. Since Ibn Battuta states that he visits a town "on the coast" and was quite clear that Mogadishu was on an island, he would seem to be talking about the settlment of Kilwa Masoko, which is on the mainland adjacent to Kilwa Kisiwani.