DNA amplification through polymerase chain reaction (PCR) has become an essential step in most DNA related procedures such as nucleic acid detection, quantification or sequencing. The increasing importance of PCR has led to a significant interest in PCR in microfluidic devices because such devices can amplify minute quantity of samples at a high throughput. While amplifying DNA strands in a continuous flow microfluidic device, samples are subjected to cyclic changes in temperature. The time-periodic temperature oscillations drive both axial and lateral velocities due to the thermal expansion of the carrier fluid, and these periodic velocity profiles results in an increase of the DNA samples dispersion. This paper investigates the Taylor dispersion of the DNA molecules undergoing reaction along with pressure driven flow in microchannels with temporally changing temperatures. We use the method of multiple time scales with regular expansions to obtain the Taylor dispersivity. We restrict the analysis to the case when the amplitude of the temperature changes is small. The cyclic variations lead to both axial and lateral velocities due to the thermal expansion of the carrier fluid. These periodic velocity profiles lead to an enhancement of the dispersion. We will report the mean velocity and the dispersion coefficients for three cases: (i) sinusoidal temperature variations with no reaction; (ii) arbitrary temporal temperature changes without reaction; and (iii) sinusoidal temperature changes with reaction. Furthermore, development of a novel idea for continuous and high throughput polymerase chain reaction on a microfluidic chip without an imposed pressure driven flow will be described. In this novel device the expansion and the contraction phases of the temperature cycling are utilized to eject and inject the sample plug, respectively. Elimination of the pressure driven flow reduces dispersion and can yield a higher throughput.