Single crystals of potassium titanyl phosphate (KTiOPO4, KTP) family (MTiOXO4, where M is K, Rb, or Cs, and X is P or As) with periodical domain structures have emerged as one of the key platforms for enabling nonlinear photonics applications. Potassium titanyl arsenate (KTiOAsO4, KTA) crystals possess nonlinear optical properties outperforming those of KTP. However, domain kinetics in KTA, being the crucial element for periodical poling, lacks comprehensive studies. We present the results of in situ imaging of domain kinetics in KTA with high temporal resolution. The analysis of a set of instantaneous domain structure images (kinetic map) has allowed reliable revealing of the slow and fast domain walls, similar to KTP. The mobility and the threshold fields for the domain walls have been estimated. The main stages of the domain structure evolution have been revealed. The original hatching stage representing the formation of quasiperiodic structure of the narrow stripe domains has been discovered. The relative input of the hatching stage has increased with external field. The obtained qualitative difference in the domain structure evolution, compared with KTP, has been attributed to a six times larger ratio of fast to slow wall mobility in KTA. This fact results in suppression of the undesirable broadening of the stripe domains thus making KTA crystals very attractive for periodical poling.