The Laplace transform was used to solve the linearized nonstationary system of equations in the Boussinesq approximation for a viscous stratified fluid in a channel with moving walls or in a vessel with a slowly moving side. The solution is analytical over all physical parameters. The boundary flow splits into the viscous and density boundary layers; their squared thicknesses are proportional to the Schmidt number. The evolution of a salt discharge is analyzed, as well as the behavior of dynamic and baroclinic vorticity components. The applicability condition for the solution is found. The solution has no stationary limit.