Counterfactual instances offer human-interpretable insight into the local behaviour of machine learning models. We propose a general framework to generate sparse, in-distribution counterfactual model explanations which match a desired target prediction with a conditional generative model, allowing batches of counterfactual instances to be generated with a single forward pass. The method is flexible with respect to the type of generative model used as well as the task of the underlying predictive model. This allows straightforward application of the framework to different modalities such as images, time series or tabular data as well as generative model paradigms such as GANs or autoencoders and predictive tasks like classification or regression. We illustrate the effectiveness of our method on image (CelebA), time series (ECG) and mixed-type tabular (Adult Census) data.