General solution representations for velocity and pressure fields describing transient flows at small Reynolds numbers (Stokes flows) and flows obeying Brinkman models are presented. The geometry dependent vector representations emerge from the incompressibility condition and are expressed in terms of just two scalar functions similar to the Papkovich-Neuber and Boussinesq-Galerkin solution type. We provide new formulae connecting our differential representations and other solutions describing unsteady Stokes flow including Lamb's (1932) general infinite series solution. The unified approach presented here further demonstrates an important link between oscillatory flows and flow through porous media using Brinkman models. It is shown that the solutions of boundary value problems in the latter can be obtained in a straightforward fashion; from the results of the former. This simple but surprising analogy is further explained using the properties (mathematical as well as physical) that are shared by the two different models. The construction of certain physical quantities is also illustrated for spherical and spheroidal inclusions. It is believed that the general solutions presented here will be useful in the computation of multi-particle interactions in transient and Brinkman flows and also in linear elasticity.