We study the complexity of the destructive bribery problem (an external agent tries to prevent a disliked candidate from winning by bribery actions) in voting over combinatorial domains, where the set of candidates is the Cartesian product of several issues. This problem is related to the concept of the margin of victory of an election which constitutes a measure of robustness of the election outcome and plays an important role in the context of electronic voting. In our setting, voters have conditional preferences over assignments to these issues, modelled by CP-nets. We settle the complexity of all combinations of this problem based on distinctions of four voting rules, five cost schemes, three bribery actions, weighted and unweighted voters, as well as the negative and the non-negative scenario. We show that almost all of these cases are $$\mathcal {NP}$$ -complete or $$\mathcal {NP}$$ -hard for weighted votes while approximately half of the cases can be solved in polynomial time for unweighted votes.