  We investigate pseudorapidity correlations of the average transverse flow of
particles emitted in relativistic heavy-ion collisions. We employ 3+1
dimensional viscous relativistic hydrodynamics with initial conditions from the
quark Glauber Monte Carlo model to confront the recent measurements on the
pseudorapidity correlations of the transverse momentum fluctuations in Pb+Pb
collisions at $\sqrt{s_{NN}}=2760$GeV. We find good agreement between the model
predictions and data. Further, we study two other observables build with the
covariance of the average transverse momentum in different rapidity bins. These
observables have better stability under various systematics, thus allowing for
a robust comparison between data and model. The transverse flow-transverse flow
correlation coefficient is directly related to correlations of the underlying
collective flow at different pseudorapidities. The 3-bin measure of $p_T$
factorization breaking in pseudorapidity gives an estimate of possible
decorrelation of the average transverse flow in the longitudinal direction.
