  The combined work of Bousquet-M\'elou, Claesson, Dukes, Jel\'inek, Kitaev,
Kubitzke and Parviainen has resulted in non-trivial bijections among ascent
sequences, (2+2)-free posets, upper-triangular integer matrices, and
pattern-avoiding permutations. To probe the finer behavior of these bijections,
we study two types of restrictions on ascent sequences. These restrictions are
motivated by our results that their images under the bijections are natural and
combinatorially significant. In addition, for one restriction, we are able to
determine the effect of poset duality on the corresponding ascent sequences,
matrices and permutations, thereby answering a question of the first author and
Parviainen in this case. The second restriction should appeal to Catalaniacs.
