  We develop the method of anholonomic frames with associated nonlinear
connection (in brief, N--connection) structure and show explicitly how
geometries with local anisotropy (various type of
Finsler--Lagrange--Cartan--Hamilton geometry) can be modeled in the
metric--affine spaces. There are formulated the criteria when such generalized
Finsler metrics are effectively induced in the Einstein, teleparallel,
Riemann--Cartan and metric--affine gravity. We argue that every generic
off--diagonal metric (which can not be diagonalized by coordinate transforms)
is related to specific N--connection configurations. We elaborate the concept
of generalized Finsler--affine geometry for spaces provided with arbitrary
N--connection, metric and linear connection structures and characterized by
gravitational field strengths, i. e. by nontrivial N--connection curvature,
Riemannian curvature, torsion and nonmetricity. We apply a irreducible
decomposition techniques (in our case with additional N--connection splitting)
and study the dynamics of metric--affine gravity fields generating Finsler like
configurations. The classification of basic eleven classes of metric--affine
spaces with generic local anisotropy is presented.
