Each of several speed-limited planar robots is driven by the acceleration, limited in magnitude. There is an unpredictable dynamic complex object, for example, a group of moving targets or an extended moving and deforming body. The robots should reach and then repeatedly trace a certain object-dependent moving and deforming curve that encircles the object and also achieve an effective self-deployment over it. This may be, for example, the locus of points at a desired mean distance or distance from a group of targets or a single extended object, respectively. Every robot has access to the nearest point of the curve and its own velocity and “sees” the objects within a finite sensing range. The robots have no communication facilities, cannot differentiate the peers, and are to be driven by a common law. Necessary conditions for the solvability of the mission are established. Under their slight and partly unavoidable enhancement, a new decentralized control strategy is proposed and shown to solve the mission, while excluding inter-robot collisions, and for the case of a steady curve, to evenly distribute the robots over the curve and to ensure a prespecified speed of their motion over it. These are justified via rigorous global convergence results and confirmed via computer simulations.