This work is devoted to the study of a class of linear timeinhomogeneous evolution equations in a scale of Banach spaces. Existence, uniqueness and stability for classical solutions is provided. We study also the associated dual Cauchy problem for which we prove uniqueness in the dual scale of Banach spaces. The results are applied to an infinite system of ordinary differential equations but also to the Fokker-Planck equation associated with the spatial logistic model in the continuum.