Let D be an integral domain and let ? be a semistar operation on D. In this paper, we define the class of ?-quasi-going-up domains, a notion dual to the class of ?-going-down domains. It is shown that the class of ?-quasi-going-up domains is a proper subclass of ?-going-down domains and that every Prüfer-?-multiplication domain is a ?-quasi-going-up domain. Next, we prove that the ?-Nagata ring Na(D, ?), is a quasi-going-up domain if and only if D is a e?-quasi-going-up and a e?-quasi-Prüfer domain. Several new characterizations are given for ?-going-down domains. We also define the universally ?-going-down domains, and then, give new characterizations of Prüfer-?-multiplication domains.