  We consider the dynamics of maximal quantum Fisher information (MQFI) after
sudden quenches for the one-dimensional transverse-field Ising model. Our
results show, the same as Loschmidt echo, there is a universality for the
revival times i.e., they do not depend on the initial state and the size of the
quench and are given by integer multiples of $T_{rev} \simeq \frac{N}{2v_{max
}}$, where $N$ is the system size and $v_{max }$ is the maximal group velocity
of quasiparticles. Critically enhanced and decreased at revival and decay times
as $T_{rev} \equiv T_{dec} $ are characterized by quenching from the order and
disorder phases into the quantum phase transition respectively, that can be
utilized to detect the quantum critical point (QCP). In some quenches crossed
from the QCP, nonanalytic behaviors appear at some times due to the turning of
the local observable from one direction to another because of identifying the
maximum value. We name this phenomenon \textit{the dynamical MQFI transitions},
occurring at the critical times $t_c$. Interestingly, although no Fisher zero
exists in the dynamics of MQFI, the first critical time emerged from the
dynamical quantum phase transition is equal to the first time whose the
logarithm of MQFI is minimum. In addition, we unveil the long-time run of MQFI
indicates a signature of a nonequilibrium quantum phase transition at the QCP.
We also discuss the probability of arising of macroscopic superpositions in the
nonequilibrium dynamics of the system.
