  The paper is devoted to the problem of existence of propagators for an
abstract linear non-autonomous evolution Cauchy problem of hyperbolic type in
separable Banach spaces. The problem is solved using the so-called evolution
semigroup approach which reduces the existence problem for propagators to a
perturbation problem of semigroup generators. The results are specified to
abstract linear non-autonomous evolution equations in Hilbert spaces where the
assumption is made that the domains of the quadratic forms associated with the
generators are independent of time. Finally, these results are applied to
time-dependent Schr\"odinger operators with moving point interactions in 1D.
