We continue our investigation of a model lattice polymer where only local conformational changes (cooperative trans <--> cis transitions) are allowed requiring the diffusion of randomness into the interior from the ends of the chain for the process of contraction of an initially extended (all-trans) chain. Simulations of the kinetics show that in the early stages of the dynamics there is a marked tendency for the cis states entering the chain to exist largely on either odd or even numbered sites in the chain thus generating a sublattice asymmetry or order in the molecule, a pattern that is largely lost at equilibrium. A similar soluble continuum model suggests that the net amount of cis in the chain should increase with time approximately as a simple exponential with the relaxation time proportional to the square of the chain length. This suggests that if a given dynamic function is plotted versus the scaled time (time divided by the chain length squared) one will obtain a general function independent of chain length. Simulations of the kinetics of the net amount of cis in the system support this scaling precisely. The difference in the number of cis states on the odd and even sublattices approaches a limiting scaled form, but develops a maximum as the chain length increases. The kinetics of the decrease in the end-to-end distance on contraction slowly approaches a limiting form in the scaled time, developing a t(-1/2) tail as the chain length goes to infinity. This later behavior is similar to the critical slowing down in the mean-field approximation to relaxation in the three-dimensional Ising model near the critical point.