Funnels are related to the big-valley hypothesis in combinatorial fitness landscapes. It suggests that local optima are not randomly distributed but are instead clustered around the global optimum, forming a coarse-grained global structure. Multi-funnel structures emerge when more than one cluster of local optima is present, some surrounding sub-optimal solutions. These multi-funnel landscapes can be challenging to search, as the optimisation process may get trapped in a sub-optimal funnel. We propose a characterisation of funnels in multi-objective combinatorial landscapes based on the solution ranks using non-dominated sorting, and a variation of the recent graph model of multi-objective landscapes: the compressed Pareto local optimal solution network (C-PLOS-net). Using a set of mnk-landscapes, we construct and visualise monotonic C-PLOS-nets, and introduce a set of metrics to characterise the landscapes’ funnel structure. The proposed metrics are found to capture the landscape global structure, to correlate with benchmark parameters, and to explain the performance of well-established multi-objective local search and evolutionary algorithms.