Gene duplication is a potential source of innovation, but the evolutionary dynamics of functional change are still poorly understood. The “ortholog conjecture” posits that such innovation or other functional change is more common after duplication than in its absence, but is still debated. We have applied a Lévy model of evolutionary trait jumps to two phenotypes of gene function: average gene expression levels, and tissue specificity. Such instantaneous jumps in trait values typically occur during the transition of organisms into new adaptive zones, and thus provide insights into the role of duplication in adaptation. Using time-calibrated phylogenies of 15 vertebrates, including 5 teleost fishes, we show for the first time that such a mechanism of trait divergence strongly affects paralogs, in addition to other modes of functional evolution. We find that at least 25% of teleost fish small-scale duplicates follow a rapid evolutionary rate shift model for both traits, much more than after speciations. We show that such trait jumps play a major role in the evolution of duplicates. While there is some evidence for more positive selection at the protein-coding level after duplication, this does not appear strongly linked to the presence of jumps in expression. This helps to explain higher phylogenetic independent contrasts between paralogs, supporting the ortholog conjecture. However, genome-wide duplicates (ohnologs) do not support such a trait jump model, and thus follow a different evolutionary dynamic, while also supporting the ortholog conjecture.