Dial-a-Ride problems consist of a setVofnvertices in a metric space (denoting travel time between vertices) and a set ofmobjects represented as source-destination pairs {(si,ti)}mi=1, where each object requires to be moved from its source to destination vertex. In themulti-vehicle Dial-a-Rideproblem, there areqvehicles, each having capacitykand where each vehiclej∈ [q] has its own depot-vertexrj∈ V. A feasible schedule consists of a capacitated route for each vehicle (where vehiclejoriginates and ends at its depotrj) that together move all objects from their sources to destinations. The objective is to find a feasible schedule that minimizes the maximum completion time (i.e.,makespan) of vehicles, where the completion time of vehiclejis the time when it returns to its depotrjat the end of its route. We study thepreemptiveversion of multi-vehicle Dial-a-Ride, in which an object may be left at intermediate vertices and transported by more than one vehicle, while being moved from source to destination. Our main results are anO(log3n)-approximation algorithm forpreemptive multi-vehicle Dial-a-Ride, and an improvedO(logt)-approximation for its special case when there is no capacity constraint (heret≤nis the number of distinct depot-vertices). There is an Ω (log1/4-ϵn) hardness of approximation known even for single vehicle capacitated Dial-a-Ride [Gørtz 2006]. For uncapacitated multi-vehicle Dial-a-Ride, we show that there are instances when natural lower bounds (used in our algorithm) are ˜Ω(logt) factor away from the optimum.We also consider the special class of metrics induced by graphs excluding any fixed minor (e.g., planar metrics). In this case, we obtain improved guarantees ofO(log2n) for capacitated multi-vehicle Dial-a-Ride, andO(1) for the uncapacitated problem.