  We establish the notion of limit consistency as a modular part in proving the
consistency of lattice Boltzmann equations (LBEs) with respect to a given
partial differential equation (PDE) system. The incompressible Navier--Stokes
equations (NSE) are used as paragon. Based upon the hydrodynamic limit of the
Bhatnagar--Gross--Krook (BGK) Boltzmann equation towards the NSE, we provide a
successive discretization by nesting conventional Taylor expansions and finite
differences. We track the discretization state of the domain for the particle
distribution functions and measure truncation errors at all levels within the
derivation procedure. Via parametrizing equations and proving the limit
consistency of the respective families of equations, we retain the path towards
the targeted PDE at each step of discretization, i.e. for the discrete velocity
BGK Boltzmann equations and the space-time discretized LBEs. As a direct
result, we unfold the discretization technique of lattice Boltzmann methods as
chaining finite differences and provide a generic top-down derivation of the
numerical scheme which upholds the continuous limit.
