  We develop variation formulas for the quantities of extrinsic geometry for
adapted variations of metrics on almost-product (e.g. foliated) Riemannian
manifolds, and apply them to study the total mixed scalar curvature of a
distribution -- analogue of the classical Einstein-Hilbert action. The mixed
scalar curvature ${\rm S}_{\,\rm mix}$ is the averaged sectional curvature over
all planes that contain vectors from both distributions of an almost-product
structure and the variations we consider preserve orthogonality of the
distributions. We derive the directional derivative $D J_{\,\rm mix}$ (of the
total ${\rm S}_{\,\rm mix}$) for adapted variations of metrics on closed
almost-product manifolds and foliations of arbitrary dimension. The obtained
Euler-Lagrange equations are presented in two equiva\-lent forms: in terms of
extrinsic geometry and intrinsically using the partial Ricci tensor. Certainly,
these mixed field equations admit amount of solutions (e.g., twisted products).
