Dissipative solitons in parametrically driven systems propagating in a spatial inhomogeneous medium are investigated. Recently, a family of dissipative solitons with an unexpected shell-type phase structure has been reported. In the present work, we show that the phase configuration moves rigidly along with the modulus after some transient state. Such a transient state is characterized for a self-adaptation of the phase front symmetry and its relative distance to the soliton. The described dynamical behavior is analytically predicted, showing good agreement with numerical simulations. A mechanism of control and manipulation of these structures based on spatial inhomogeneities is proposed.