We give some necessary and sufficient conditions for (global) continuity of the limit of a pointwise convergent net of cone metric space-valued functions, defined on a Hausdorff topological space, in terms of weak filter exhaustiveness. In this framework, we prove some Ascoli-type theorems, considering also possibly asymmetric and extended real-valued distance functions. Furthermore, we pose some open problems. MSC: Primary 26E50; 28A12; 28A33; 28B10; 28B15; 40A35; 46G10; 54A20; 54A40; secondary 06F15; 06F20; 06F30; 22A10; 28A05; 40G15; 46G12; 54H11; 54H12; 47H10