This is an issue of some scholarly debate. We are certain that there was extensive cultural interchange, including trade, treaties, and letters (once writing was invented). The problem with identifying "diffusion" is in determining whether, for example, writing was invented in Mesopotamia and diffused to Egypt, vice versa, or whether it was independently invented in both regions. This requires dating more precise than is actually possible. 

One of the cases where we can make a precise and specific claim of diffusion is that of the manufacture of the so-called "Egyptian Faience," which was actually first developed in Mesopotamia, then Phoenicia, but elaborated and perfected in Egypt. 

Faience is a type of glazed ceramic with a nearly glassy texture. Many of the most dramatic examples from Egypt are of a cobalt-blue color with gold flecks. 

Since Mesopotamia is widely regarded as the "crucible of civilization" there are many different examples of cultural diffusion that can be pointed out when looking at Egyptian culture.

Environmentally the Mesopotamians developed many ways of changing the landscape of the desert to support their way of life. The practices of irrigation to bring river water to crops began on the Tigirs and Euphartes and eventually made its way to the Nile Delta. The practices of mining and animal domestication also began in Mesopotamia.

Culturally many important practices also diffused to Egypt where they had their own twist put on them. Egyptian hieroglyphics developed from figures that were similar to Mesopotamian cunieform. Social structures and religious obligations were also taken and improved upon. The very famous ceremonial burial rights that are associated with Egypt actually made there way there from the city-states of Mesopotamia. 

Economical practices were also tranferred between the two empires. The importance of the market and agricultural products became mainstays of both kingdoms. The social sturcture which was supported by the economy was also the result of cultural diffusion.