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The existence of distributional chaos and distributional irregular vectors has been recently considered in the study of linear dynamics of operators and C0-semigroups. In this paper we extend some previous results on both notions to sequences of operators, C0-semigroups, C-regularized semigroups, and α-times integrated semigroups on Fréchet spaces. We also add a study of rescaled distributionally chaotic C0-semigroups. Some examples are provided to illustrate all these results.The first and fourth authors are supported in part by MEC Project MTM2010-14909, MTM2013-47093-P, andPrograma de Investigaci´on y Desarrollo de la UPV, Ref. SP20120700. The second author is partially supported bygrant 174024 of Ministry of Science and Technological Development, Republic of Serbia. The third author has beenpartially supported by Project MTM2013-42105-P, DGI-FEDER, of the MCYTS; Project E-64, D.G. Arag´on, andProject UZCUD2014-CIE-09, Universidad de Zaragoza. The forth author is supported by a grant of the FPU Programof Ministry of education of Spain

Distributionally chaotic families of operators on Fréchet spaces