  We consider the static and dynamic phases in a Rosenzweig-Porter (RP) random
matrix ensemble with the tailed distribution of off-diagonal matrix elements of
the form of the large-deviation ansatz. We present a general theory of survival
probability in such a random-matrix model and show that the {\it averaged}
survival probability may decay with time as the simple exponent, as the
stretch-exponent and as a power-law or slower. Correspondingly, we identify the
exponential, the stretch-exponential and the frozen-dynamics phases. As an
example, we consider the mapping of the Anderson model on Random Regular Graph
(RRG) onto the "multifractal" RP model and find exact values of the
stretch-exponent $\kappa$ depending on box-distributed disorder in the
thermodynamic limit. As another example we consider the logarithmically-normal
RP (LN-RP) random matrix ensemble and find analytically its phase diagram and
the exponent $\kappa$. In addition, our theory allows to compute the shift of
apparent phase transition lines at a finite system size and show that in the
case of RP associated with RRG and LN-RP with the same symmetry of distribution
function of hopping, a finite-size multifractal "phase" emerges near the
tricritical point which is also the point of localization transition.
