Leadership is commonly invoked when accounting for the coordination of group movements in animals, yet it remains loosely defined. In parallel, there is increased evidence of the sharing of group decisions by animals on the move. How leadership integrates within this recent framework on collective decision-making is unclear. Here, we question the occurrence of leadership in horses, a species in which this concept is of prevalent use. The relevance of the three main definitions of leadership--departing first, walking in front travel position, and eliciting the joining of mates--was tested on the collective movements of two semi-free ranging groups of Przewalski horses (Equus ferus przewalskii). We did not find any leader capable of driving most group movements or recruiting mates more quickly than others. Several group members often displayed pre-departure behaviours at the same time, and the simultaneous departure of several individuals was common. We conclude that the decision-making process was shared by several group members a group movement (i.e., partially shared consensus) and that the leadership concept did not help to depict individual departure and leading behaviour across movements in both study groups. Rather, the different proxies of leadership produced conflicting information about individual contributions to group coordination. This study discusses the implications of these findings for the field of coordination and decision-making research.