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We study the evolution of passive scalar fields that are maintained by steady but spatially inhomogeneous sources and sinks, and stirred by statistically stationary homogeneous and isotropic incompressible flows including "familiar" turbulence. The effectiveness of a flow field to enhance mixing over molecular diffusion is measured by the suppression of the spacetime averaged scalar variance, gradient variance (stressing small length scales), and inverse gradient variance (focusing on large scale fluctuations). Ratios of these variances without stirring to the corresponding variances with stirring provide nondimensional measures of the "mixing efficiency" at different length scales. In this work we derive rigorous bounds of these multiscale mixing efficiencies as a function of the Peclet number, the natural gauge of the intensity of the stirring. For a wide variety of sources and sinks the upper estimates for all the efficiencies scale (with the Peclet number) the same as predicted by a classical eddy diffusion model. In some cases we can show that there are flows that saturate the bounds. However for some other sourcesink configurations stirred by statistically stationary homogeneous isotropic flows, the efficiency estimates at different length scales display distinct and "anomalous" subclassical scaling with respect to the Peclet number. Direct numerical simulations, exact and asymptotic calculations for some model problems are presented to illustrate the phenomena. Host: Robert E. Ecke 