Hey, sorry for not getting to this question earlier.

So yeah, there was a huge amount of influence on Swahili culture coming from Yemen, Iraq, Iran, India and China.

That influence can be felt in Swahili vocabulary, where words for foods like Areca Nut, Betel nut, cotton or words like steel and rope all have origins from Persian. Word for Pigeon Pea comes from India and the plant originates in India, while the word for "unripe coconu" seems to be related to Hindi. The word for Yam comes from Tamil or perhaps via Malay. Of course, the Swahili language is fundamentally a Bantu language, and is Bantu in its grammar. So, there was influence from East Africa.

There was a great deal of religious influence from the middle east, with Mosques containing Arabic inscriptions in the Kufic script (which was developed in Iraq). In the 1000s to early 1300s there is evidence of Sunnis, Shiites and Kharijite-Ibadis all present. In the 1300s and 1400s, greater influence from Egypt and Yemen means a solidification of Sunni influence, although Sufi saints continue to be venerated.

Archaeologically, there is lots of evidence of beads originating from India (particularly from Gujarat) which arrive on the coast of East Africa. This evidence is also supported by contemporary Arabic and Chinese sources from the 11th-15th centuries which mention Gujarati merchants bringing cotton cloth, beads, copper and porcelain to Zanzibar and the mouth of the Zambezi river.

Neville Chittick's digs at Manda island off the coast of Kenya has turned up pottery coming from China, India, Iran, and "Islamic ware" (i.e. from Arabian peninsula, the gulf or Egypt) from the entire period from 800-1500.

Coinage also shows influence from Egypt, with small silver coins following Egyptian patterns from 1000-1300.

There were also impacts on social customs. Randall Pouwels notes a custom that it was considered bad luck to transport coconuts aboard mtepe ships. He argues that this might be a reflection of folk-memory of Indian and Austronesian slave raiding along the coast, and association of Austronesian slave raiding with coconut crops they introduced.

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In terms of when trade "ramped up". We have early written evidence of Roman Egyptian trade contact with the East African coast in the first century AD, described in the Periplus of the Erythraean Sea. Also, the archaeologist Felix Chami has found Roman beads in the Rufiji delta in Tanzania.

Arabic sources from the 8th century onward speak about pre-Islamic trade and contacts with the peoples along the East African coast, which continued and intensified in the Islamic era.

The period from 1000 to the early 1300s seems to have been a zenith of trade and engagement of Swahili cities and the East African littoral with the Indian Ocean world. Political chaos in the Islamic world caused by the Mongol conquest, along with a cooling spell seems to have caused a decline in trade in the middle-fourteenth century, and a decline in population in Swahili cities (along with the decline in the Great Zimbabwe civilization).

However, after a half-century lull, there is evidence for a revival in trade and contacts in the Indian ocean from the last quarter century of the fourteenth century into the fifteenth century.

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A really good book to consult on this is The Swahili World edited by Adria LaViolette and Stephanie Wynn-Jones. Routledge, 2017. Basically all the information here is from chapter 34 by Philip Beajard.

Neville Chittick's excavations at Manda were published as Manda- Excavations at an Island port on the Kenya Coast by British Institute in East Africa in 1984.

Randall Pouwels book is Horn and Crescent, though my summary of his point is heard through Philip Bajard in The Swahili World

Edit- Also, I really elided over the whole issue of the influence of the African hinterland on the coastal communities. Jonathan Walz has a chapter (ch 36) in The Swahili World which covers "inland entanglement with the Swahili world" from the 700s to the 1550s.