Is transition more common than transversion during the evolution of duplicated genes?
If we consider that the biochemical impact of transitions vs transversions is constant across the two scenarios, then the only discrepancy can be due to the level of purifying selection, or the level of positive diversifying selection, or both, between the two duplicates. 

By far the most common fate to one duplicate copy is for purifying selection to be relaxed, leading to accruing deleterious mutations and eventually loss. This means that fixation of non-synonymous mutations are more common, and as transversions are more likely to be non-synonymous, this implies that the rate of fixation for transversions would increase. 

Similarly, one might expect functional change leading to positive diversifying selection, which implies a greater rate of fixation of non-synonymous mutations and, following the same logic as above, a greater rate of transversions. 

Whether or not this contributing increase to the rate of fixation of transversions mean that transversions actually outnumber transitions, or instead merely decreases the ratio of transition:transversion, will be dependent on the particular gene and selective scenario. 

As a final note, while the first cause I identified (biochemical)  actually changes the rate of mutation, the selection-specific factors are changes to rate of fixation, which is a population-level process. So while the relative rates of transition and transversion may stay the same, the fixation rates of these two categories of mutation between the two duplicate copies are the important difference.