This paper introduces LMC-WOD (Local Markovian Consensus with Weak Ordinal Dominance), an extension of the LMC method designed to aggregate rankings that include ties and inconsistent pairwise preferences. The approach builds a local Markov chain based on direct dominance relations and computes a stationary distribution to represent the collective consensus. Unlike global methods, LMC-WOD relies on local interactions and supports partial and weak orders. It handles transitivity violations, reflects local consensus, and remains robust to small perturbations. Two case studies illustrate its behavior under consistent and conflicting inputs, with and without ties. A comparison with PageRank shows how each method propagates influence differently across alternatives. Results confirm that LMC-WOD can capture both strong and weak ordinal patterns and highlight its potential for applications where standard ranking assumptions do not hold. Its sensitivity to highly diverse rankings and the use of a fixed damping factor highlights areas for further investigation.