Anticipating future situations from streaming sensor data is a key perception challenge for mobile robotics and automated vehicles. We address the problem of predicting the path of objects with multiple dynamic modes. The dynamics of such targets can be described by a Switching Linear Dynamical System (SLDS). However, predictions from this probabilistic model cannot anticipate when a change in dynamic mode will occur. We propose to extract various types of cues with computer vision to provide context on the target's behavior, and incorporate these in a Dynamic Bayesian Network (DBN). The DBN extends the SLDS by conditioning the mode transition probabilities on additional context states. We describe efficient online inference in this DBN for probabilistic path prediction, accounting for uncertainty in both measurements and target behavior. Our approach is illustrated on two scenarios in the Intelligent Vehicles domain concerning pedestrians and cyclists, so-called Vulnerable Road Users (VRUs). Here, context cues include the static environment of the VRU, its dynamic environment, and its observed actions. Experiments using stereo vision data from a moving vehicle demonstrate that the proposed approach results in more accurate path prediction than SLDS at the relevant short time horizon (1 s). It slightly outperforms a computationally more demanding state-of-the-art method.