Each of several planar mobile robots is driven by the acceleration vector, upper-limited in magnitude, obeys a given speed upper bound, has no communication facilities, and cannot distinguish between the peers. There is an unpredictably moving and deforming simple closed curve in the plane, e.g., the locus of points at a desired distance from the edges of a single 2D or 1D or point-wise targeted object or at a desired mean distance from a group of such objects. In its local frame, every robot “sees” the objects within a finite range of “visibility”, has access to its own velocity, and is also able to determine the nearest point on the curve. The robots should autonomously reach the curve, subsequently track it, and achieve an effective self-distribution over it. We first establish necessary conditions for the solvability of this mission. Then we propose a new navigation law and show that it does solve the mission under slight and partly unavoidable enhancement of those conditions, while excluding inter-robot collisions. For steady curves, this law also ensures an even self-distribution of the robots and a pre-specified speed of their motion over the curve. These traits are justified via rigorous global convergence results and are confirmed via computer simulation tests.