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We present an analytical formalism, supported by numerical simulations, for studying forces that act on curved walls following temperature quenches of the surrounding ideal Brownian fluid. We show that, for curved surfaces, the post-quench forces initially evolve rapidly to an extremal value, whereafter they approach their steady state value algebraically in time. In contrast to the previously-studied case of flat boundaries (lines or planes), the algebraic decay for the curved geometries depends on the dimension of the system. Specifically, the steady-state values of the force are approached in time as $t^{-d/2}$ in d-dimensional spherical (curved) geometries. For systems consisting of concentric circles or spheres, the exponent does not change for the force on the outer circle or sphere. However, the force exerted on the inner circle or sphere experiences an overshoot and, as a result, does not evolve towards the steady state in a simple algebraic manner. The extremal value of the force also depends on the dimension of the system, and originates from the curved boundaries and the fact that particles inside a sphere or circle are locally more confined, and diffuse less freely than particles outside the circle or sphere.