  Bratteli-Vershik models have been very successfully applied to the study of
various dynamical systems, in particular, in Cantor dynamics. In this paper, we
study dynamics on the path spaces of generalized Bratteli diagrams that form
models for non-compact Borel dynamical systems. Generalized Bratteli diagrams
have countably infinite many vertices at each level, thus the corresponding
incidence matrices are also countably infinite. We emphasize differences (and
similarities) between generalized and classical Bratteli diagrams. Our main
results: $(i)$ We utilize Perron-Frobenius theory for countably infinite
matrices to establish criteria for the existence and uniqueness of
tail-invariant path space measures (both probability and $\sigma$-finite).
$(ii)$ We provide criteria for the topological transitivity of the tail
equivalence relation. $(iii)$ We describe classes of stationary generalized
Bratteli diagrams (hence Borel dynamical systems) that: $(a)$ do not support a
probability tail-invariant measure, $(b)$ are not uniquely ergodic with respect
to the tail equivalence relation. $(iv)$ We describe classes of generalized
Bratteli diagrams which can or cannot admit a continuous Vershik map and
construct a Vershik map which is a minimal homeomorphism of a (non locally
compact) Polish space. $(v)$ We provide an application of the theory of
stochastic matrices to analyze diagrams with positive recurrent incidence
matrices.
