Recent technological advancements have given rise to the ability of collecting vast amounts of data, that often exceed the capacity of commonly used machine learning algorithms. Approaches such as coresets and synthetic data distillation have emerged as frameworks to generate a smaller, yet representative, set of samples for downstream training. As machine learning is increasingly applied to decision-making processes, it becomes imperative for modelers to consider and address biases in the data concerning subgroups defined by factors like race, gender, or other sensitive attributes. Current approaches focus on creating fair synthetic representative samples by optimizing local properties relative to the original samples. These methods, however, are not guaranteed to positively affect the performance or fairness of downstream learning processes. In this work, we present Fair Wasserstein Coresets (FWC), a novel coreset approach which generates fair synthetic representative samples along with sample-level weights to be used in downstream learning tasks. FWC aims to minimize the Wasserstein distance between the original datasets and the weighted synthetic samples while enforcing (an empirical version of) demographic parity, a prominent criterion for algorithmic fairness, via a linear constraint. We show that FWC can be tought of as a constrained version of Lloyd's algorithm for k-medians or k-means clustering.Our experiments, conducted on both synthetic and real datasets, demonstrate the scalability of our approach and highlight the competitive performance of FWC compared to existing fair clustering approaches, even when attempting to enhance the fairness of the latter through fair pre-processing techniques.