The generalised median string is defined as a string that has the smallest sum of distances to the elements of a given set of strings. It is a valuable tool in representing a whole set of objects by a single prototype, and has interesting applications in pattern recognition. All algorithms for computing generalised median strings known from the literature are of static nature. That is, they require all elements of the underlying set of strings to be given when the algorithm is started. In this paper, we present a novel approach that is able to operate in a dynamic environment, where there is a steady arrival of new strings belonging to the considered set. Rather than computing the median from scratch upon arrival of each new string, the proposed algorithm needs only the median of the set computed before together with the new string to compute an updated median string of the new set. Our approach is experimentally compared to a greedy algorithm and the set median using both synthetic and real data.