We suggest an experimentally feasible procedure to observe counterflow and paired superfluidity in ultracold atom systems. We study the time evolution of one-dimensional mixtures of bosonic atoms in an optical lattice following an abrupt displacement of an additional weak confining potential. We find that the dynamic responses of the paired superfluid phase for attractive interspecies interactions and the counterflow superfluid phase for repulsive interactions are qualitatively distinct and reflect the quasi-long-range order that characterizes these phases. These findings suggest a clear experimental procedure for their detection, and give an intuitive insight into their dynamics.