Studying properties of evolution equations arising in different physical contexts commonly starts from assuming the traveling wave (TW) solution form which reduces the problem to an ordinary differential equation (ODE). A variety of direct methods for finding such solutions have been designed but usually there is no algorithmic way to proceed further from this stage. In the present study, a method, which allows constructing non-traveling wave solutions of an evolution equation from known traveling wave solutions, is developed and applied to some types of equations. The transformations yielded by the method can be naturally used for finding new solutions of a given equation. Having the TW solutions (for example, solitary wave solutions) defined in an explicit form, more general non-TW solutions can be also explicitly determined. The transformations can also give insight into some general properties of the equations.