  We study the variational principle and derivation of the field equations for
different classes of teleparallel gravity theories, using both their
metric-affine and covariant tetrad formulations. These theories have in common
that in addition to the tetrad or metric, they employ a flat connection as
additional field variable, but differ by the presence of absence of torsion and
nonmetricity for this independent connection. Besides the different underlying
geometric formulation using a tetrad or metric as fundamental field variable,
one has different choices to introduce the conditions of vanishing curvature,
torsion and nonmetricity, either by imposing them a priori and correspondingly
restricting the variation of the action when the field equations are derived,
or by using Lagrange multipliers. Special care must be taken, since these
conditions form non-holonomic constraints. Here we show explicitly that all of
the aforementioned approaches are equivalent, and that the same set of field
equations is obtained, independently of the choice of the geometric formulation
and variation procedure. We further discuss consequences arising from the
diffeomorphism invariance of the gravitational action, and show how they
establish relations between the gravitational field equations.
