This paper concerns channel tracking in a multiantenna context for correlated flat-fading channels obeying a Gauss-Markov model. It is known that data-aided tracking of fast-fading channels requires a lot of pilot symbols in order to achieve sufficient accuracy, and hence decreases the spectral efficiency. To overcome this problem, we design a code-aided estimation scheme which exploits information from both the pilot symbols and the unknown coded data symbols. The algorithm is derived based on a factor graph representation of the system and application of the sum-product algorithm. The sum-product algorithm reveals how soft information from the decoder should be exploited for the purpose of estimation and how the information bits can be detected. Simulation results illustrate the effectiveness of our approach.