This work establishes rigorous, novel and widely applicable stability guarantees and transferability bounds for general graph convolutional networks -- without reference to any underlying limit object or statistical distribution. Crucially, utilized graph-shift operators are not necessarily assumed to be normal, allowing for the treatment of networks on both directed- and undirected graphs within the developed framework. In the undirected setting, stability to node-level perturbations is related to an 'adequate spectral covering' property of the filters in each layer. Stability to edge-level perturbations is discussed and related to properties of the utilized filters such as their Lipschitz constants. Results on stability to vertex-set non-preserving perturbations are obtained by utilizing recently developed mathematical-physics based tools. As an exemplifying application of the developed theory, it is showcased thatgeneral graph convolutional networks utilizing the un-normalized graph Laplacian as graph-shift-operator can be rendered stable to collapsing strong edges in the underlying graph if filters are mandated to be constant at infinity. These theoretical results are supported by corresponding numerical investigations showcasing the response of filters and networks to such perturbations.