We prove certain rigidity properties of higher-rank abelian product actions of the type \alpha\times \mathop{\rm Id}\nolimits_N:{\mathbb Z}^\kappa\to\operatorname{Diff}(M\times N), where \alpha is (TNS) (i.e. is hyperbolic and has some special structure of its stable distributions). Together with a result about product actions of property (T) groups, this implies the local rigidity of higher-rank lattice actions of the form \alpha\times \mathop{\rm Id}\nolimits_{\mathbb{T}}:\Gamma\to\operatorname{Diff}(M\times\mathbb{T}), provided \alpha has some rigidity properties itself, and contains a (TNS) subaction.