P. Das et al. recently introduced and studied the notions of strong AI-summability with respect to an Orlicz function F and AI-statistical convergence, where A is a nonnegative regular matrix and I is an ideal on the set of natural numbers. In this paper, we will generalise these notions by replacing A with a family of matrices and F with a family of Orlicz functions or moduli and study the thus obtained convergence methods. We will also give an application in Banach space theory, presenting a generalisation of Simons' sup-limsup-theorem to the newly introduced convergence methods (for the case that the filter generated by the ideal I has a countable base), continuing some of the author's previous work.