A special kicked rotor moving in a piecewise continuous force field is suggested and studied. When adjusting a control parameter,the rotor displays a transition from conservative to quasi-dissipative, and a change of a stochastic web formed by the set of the images of the discontinuous borderlines to a transient stochastic web. This transition and change can be described by a logarithmic rule of the fractal dimension of the transient stochastic web versus the control parameter, or a rule of the averaged lifetime of the iterations in the transient stochastic web versus the control parameter.