Throughout history we are aware of cultural exchange via descriptions of foreign culture and through the arts. This is a particularly easy concept to convey, today, with the extensive globalization that has occurred through trade and technology. Many folks, for example, are already familiar with the relatively new music genre of reggaeton, a blend of Caribbean (especially Jamaican) dance hall music, Latin American salsa, and Latin hip hop that grew out of American rap traditions.
The past offers many examples of architectural and artistic transmission from one culture to the next, sometimes revealing relations that were otherwise unknown to scholars. This is, therefore, not a strictly historical area of research, as most examples are of architecture or artifacts–and these examples date back well before the historical record was established, providing revelations into the development of the earliest human cultures before permanent settlements were established.
Another area with copious evidence is in culinary culture. We also see it in food, today, just as the Italians witnessed it with the introduction of Chinese and other Far Eastern noodles, creating pasta. (The spice trade has been longtime indicator of cultural exchange, and the Silk Road has revealed many secrets of cultural development and transmission. It was along the route that some of the best evidence of the secret cult of the Manicheans was finally revealed, as opposed to the meager evidence within the realm of the Roman Empire where it was founded!) So, foodways are another legitimate way to pursue this same idea. (See Colonial Foodways, A delicious learning experience, for example.)
This idea was recently reinforced when I came across a fascinating YouTube video produced by the Museum of Russian Icons in Massachusetts via Twitter. In it the museum explains with digital animation how a Russian icon is made. What really makes this interesting is the knowledge that both Russian Orthodoxy and its icons come from Byzantine (or Greek) Orthodoxy. For that matter, Cyrillic, the written language of Russia and parts of eastern Europe is the likely heir of a written language invented by Byzantine scholars Cyril and Methodious and derived from Greek in the mid-9th century (precise origins of Cyrillic and its inventor/s is under some dispute by scholars, today).
These sorts of cultural exchanges are richly represented in history through artifacts and historical commentary. It is an inviting and exciting way to study history, especially when one recognizes one culture while studying another.