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In this paper, by introducing a new type asymptotic coupling by reflection,we explore the long time behavior of random probability measure flowsassociated with a large class of one-dimensional McKean-Vlasov SDEs with commonnoise. Concerning the McKean-Vlasov SDEs with common noise under considerationin the present work, in contrast to the existing literature, the drift termsare much more general rather than of the convolution form, and, in particular,can be of polynomial growth with respect to the spatial variables, and moreoveridiosyncratic noises are allowed to be of multiplicative type. Mostimportantly, our main result indicates that both the common noise and theidiosyncratic noise facilitate the exponential contractivity of the associatedmeasure-valued processes.

Long time behavior of one-dimensional McKean-Vlasov SDEs with common noise