The prompt emission phase of gamma-ray bursts (GRBs) exhibits two distinct patterns of the peak-energy (E p ) evolution; i.e., time-resolved spectral analyses of νF ν spectra of broad pulses reveal (1) "hard-to-soft" and (2) "flux-tracking" patterns of E p evolution in time, the physical origin of which still remains not well understood. We show here that these two patterns can be successfully reproduced within a simple physical model invoking synchrotron radiation in a bulk-accelerating emission region. We show further that the evolution patterns of the peak energy have, in fact, direct connections to the existence of two different (positive or negative) types of spectral lags, seen in the broad pulses. In particular, we predict that (1) only the positive type of spectral lags is possible for the hard-to-soft evolution of the peak energy, (2) both the positive and negative type of spectral lags can occur in the case of flux-tracking pattern of the peak energy, (3) for the flux-tracking pattern, the peak location of the flux light curve slightly lags behind the peak of the E p evolution with time if the spectral lags are positive, and (4) in the case of flux-tracking pattern, double-peaked broad pulses can appear in the light curves, the shape of which is energy-dependent.