In this paper, we consider the problem of testing the equality of two multivariate distributions based on geometric graphs constructed using the interpoint distances between the observations. These include the tests based on the minimum spanning tree and the $K$-nearest neighbor (NN) graphs, among others. These tests are asymptotically distribution-free, universally consistent and computationally efficient, making them particularly useful in modern applications. However, very little is known about the power properties of these tests. In this paper, using the theory of stabilizing geometric graphs, we derive the asymptotic distribution of these tests under general alternatives, in the Poissonized setting. Using this, the detection threshold and the limiting local power of the test based on the $K$-NN graph are obtained, where interesting exponents depending on dimension emerge. This provides a way to compare and justify the performance of these tests in different examples.