We describe the asymptotic behavior of solutions of the Cauchy problem for the linear differential equation of the first order in the Banach space, containing a small paremeter before the derivative. Denote the main variable by t and the generator of the equation by -iA(t). It is supposed that the linear operator A(t) for any t on a given finite interval has continuous spectrum (plus, may be, some number of eigenvalues) and allows the reduction to a constant model operator by application of methods of the stationary scattering theory. For example, A(t) can be a Schroedinger operator with a quickly decreasing at infinity potential depending on t, such that the initial equation is a non-stationary Schroedinger equation. Then the asymptotic behavior of solutions of the problem can be described by formulas that are similar to the known formulas for the case of the generator with purely point simple (for any t) spectrum when the ideas of the "Quantum Adiabatic Theorem" can be used. In particular, that allows to describe the asymptotic behavior of the solution whose initial data at t=0 is the eigenfunction of the continuous spectrum of A(0).