We study diffusion-limited coalescence, A+A⇋A, in one dimension, in the presence of a diffusing trap. The system may be regarded as a generalization of von Smoluchowski's model for reaction rates, in that (a) it includes reactions between the particles surrounding the trap, and (b) the trap is mobile—two considerations which render the model more physically relevant. As seen from the trap's frame of reference, the motion of the particles is highly correlated, because of the motion of the trap. An exact description of the long-time asymptotic limit is found using the IPDF method and exploiting a “shielding” property of reversible coalescence that was discovered recently. In the case where the trap also acts as a sources—giving birth to particles—the shielding property breaks down, but we find an “equivalence principle”: Trapping and diffusion of the trap may be compensated by an appropriate rate of birth, such that the steady state of the system is identical with the equilibrium state in the absence of a trap.