In the early days of Gabor analysis it was a common to say that Gabor expansions of signals are interesting due to the natural interpretation of Gabor coefficients, but unfortunately the computation of Gabor coefficients is costly. Nowadays a large variety of efficient numerical algorithms exists and it has been recognized that stable and robust Gabor expansions can be achieved at low redundancy, e.g., by using a Gaussian atom and any time-frequency lattice of the form (see formula in paper). Consequently Gabor multipliers, i.e., linear operators obtained by applying a pointwise multiplication of the Gabor coefficients, become an important class of time-variant filters. It is the purpose of this paper to describe that fact that - provided one uses Gabor atoms from a suitable subspace (formula in paper)one has the expected continuous dependence of Gabor multipliers on the ingredients. In particular, we will provide new results which show that a small change of lattice parameters implies only a small change of the corresponding Gabor multiplier (e.g., in the Hilbert-Schmidt norm).