We present a study of the average transverse momentum ($p_t$) fluctuations and $p_t$ correlations for charged particles produced in Cu+Cu collisions at midrapidity for $\sqrt{s_{NN}} =$ 62.4 and 200 GeV. These results are compared with those published for Au+Au collisions at same energies, to explore the system size dependence. In addition to the collision energy and system size dependence, the $p_t$ correlations results have been studied as functions of the collision centralities, the ranges in $p_t$, the pseudo-rapidity $\eta$, and the azimuthal angle $\phi$, for which the correlations are measured. The square root of the measured $p_t$ correlations when scaled by mean-$p_t$ are found to be independent of both colliding beam energy and system size studied. The transport based model calculations are found to have a better quantitative agreement with the measurements compared to models which incorporate only jet-like correlations.