The aim of this paper is to push a step towards the development of a surrogate-assisted methodology for expensive optimization problems that have both a combinatorial and a multiobjective nature. We target pseudo-boolean multiobjective functions, and we provide a comprehensive study on the design of a modular framework integrating three main configurable components. The proposed framework is based on the Walsh basis as a surrogate, and on a decomposition-based evolutionary paradigm for maintaining the solution set. The three considered components are: (i) the inner optimizer used for handling the soconstructed Walsh surrogate, (ii) the selection strategy allowing to decide which solution is to be evaluated by the expensive objectives, and (iii) the strategy used to setup the Walsh order hyper-parameter. Based on a thorough empirical analysis relying on two benchmark problems, namely bi-objective NK-landscapes and UBQP problems, we show the effectiveness of the proposed framework with respect to the available budget in terms of calls to the evaluation function. More importantly, our empirical findings shed more lights on the combined effects of the investigated components on search performance, thus providing a better understanding of the key challenges for designing a successful surrogate-assisted multiobjective combinatorial search process.