  We present generic form of the covariant nonlocal action for infrared
modifications of Einstein theory recently suggested within the weak-field
curvature expansion. In the lowest order it is determined by two nonlocal
operators -- kernels of Ricci tensor and scalar quadratic forms. In models with
a low strong-coupling scale this action also incorporates the strongly coupled
mode which cannot be perturbatively integrated out in terms of the metric
field. This mode enters the action as a Lagrange multiplier for the constraint
on metric variables which reduces to the curvature scalar and enforces the
latter to vanish on shell. Generic structure of the action is demonstrated on
the examples of the Fierz-Pauli and Dvali-Gabadadze-Porrati models and their
extensions. The gauge-dependence status of the strong-coupling and VDVZ
problems is briefly discussed along with the manifestly gauge-invariant
formalism of handling the braneworld gravitational models at classical and
quantum levels.
