Generally, the Gauge theories have as a geometrical support the associated bundles of a principal G-bundle. An associate bundle of a $k$-principal frame bundle $\mathrm P^k(\mathrm M)$ is the $k$-osculator bundle, $\operatorname{Osc}^k\mathrm M$ over some manifold M [8]. In the paper [9], we defined and studied the general concepts concerning the higher order gauge transformations. The theory becomes consistent when investigating the Gauge invariance of $k$-order Lagrangians. In the present paper we shall study the global and local invariance of $k$-order Lagrangians under the Gauge infinitesimal transformations from classical Gauge theory, generalizing the result from [3].