For thermal management architectures wherein the heat sink is embedded close to a dynamic heat source, non-uniformities may propagate through the heat sink base to the coolant. Available transient models predict the effective heat spreading resistance to calculate chip temperature rise, or simplify to a representative axisymmetric geometry. The coolant-side temperature response is seldom considered, despite the potential influence on flow distribution and stability in two-phase microchannel heat sinks. This study solves three-dimensional transient heat conduction in a Cartesian chip-on-substrate geometry to predict spatial and temporal variations of temperature on the coolant side. The solution for the unit step response of the three-dimensional system is extended to any arbitrary temporal heat input using Duhamel's method. For time-periodic heat inputs, the steady-periodic solution is calculated using the method of complex temperature. As an example case, the solution of the coolant-side temperature response in the presence of different transient heat inputs from multiple heat sources is demonstrated. To represent a case where the thermal spreading from a heat source is localized, the problem is simplified to a single heat source at the center of the domain. Metrics are developed to quantify the degree of spatial and temporal non-uniformity in the coolant-side temperature profiles. These non-uniformities are mapped as a function of nondimensional geometric parameters and boundary conditions. Several case studies are presented to demonstrate the utility of such maps.