  Impurities and defects in Kondo insulators can have an unusual impact on
dynamics that blends with effects of intrinsic electron correlations. Such
crystal imperfections are difficult to avoid, and their consequences are
incompletely understood. Here we study magnetic impurities in Kondo insulators
via perturbation theory of the s-d Kondo impurity model adapted to small
bandgap insulators. The calculated magnetization and specific heat agree with
recent thermodynamic measurements in samarium hexaboride (SmB$_6$). This
qualitative agreement supports the physical picture of multi-channel Kondo
screening of local moments by electrons and holes involving both intrinsic and
impurity bands. Specific heat is thermally activated in zero field by Kondo
screening through sub-gap impurity bands and exhibits a characteristic upturn
as the temperature is decreased. In contrast, magnetization obtains a dominant
quantum correction from partial screening by virtual particle-hole pairs in
intrinsic bands. We point out that magnetic impurities could impact de Haas-van
Alphen quantum oscillations in SmB$_6$, through the effects of Landau
quantization in intrinsic bands on the Kondo screening of impurity moments.
