We consider both analytically (by the supersymmetric non-linear sigma-model) and numerically (by exact diagonalization) the local density of states parametric correlation function in the multifractal phase of the Rosenzweig-Porter random matrix model. The results show that the mini-band in the local spectrum is shifted coherently without its destruction as the parameters of the Hamiltonian are varied. Since the number of levels in the mini-band is extensive with the size of the matrix such a collective response greatly enhances sensitivity of the system to adiabatic perturbation. This effect is expected to hold in the Anderson model on random regular graph (where the number of levels in a mini-band is finite, albeit parametrically large close to localization transition) and in realistic many-body systems.