We prove local convergence results for the spectra and pseudospectra of sequences of linear operators acting in different Hilbert spaces and converging in generalised strong resolvent sense to an operator with possibly non-empty essential spectrum. We establish local spectral exactness outside the limiting essential spectrum, local \varepsilon -pseudospectral exactness outside the limiting essential \varepsilon -near spectrum, and discuss properties of these two notions including perturbation results.