No change should be observed as far as your crucible is hermetically closed. Extreme cases sould be that your flow rate is so high that it cools the crucible, or that the crucible is not very well closed and the gas is impure, either because it is intrinsically so or because of a leak in the device. In the latter case, irreversible shift of the Tg might be seen due to oxidation favoured by a higher flow rate. If these extreme situations don't occur, the Tg should be stable.

You should check the repeatability of your measurements, by doing several cycles, before changing the flow rate and see what happens.

Alain

A change of 20 ml/min in flow rate will barely affect the glass transition temperature if you use nitrogen as a purge gas. When using gases with higher thermal conductivity such as helium, relatively small changes in the purge gas flow rate may have clear effects on the measured reaction enthalpy. Ideally the flow rate of the purge gas is kept constant and monitored by a gas flow controller. If a significant change in purge gas flow rate is needed a new instrument calibration using the same purge gas flow rate should be performed. As already mentioned, effects of variable purge rates are by far less critical for nitrogen than for helium. 

Just a short additional comment: To ensure that the temperature is measured correctly in DSC measurements, you should perform the calibration with the same type of purge gas and the same flow rate as in the subsequent experiment.

Purge gas flow will not alter the transition temperatures. Calibrate the DSC for transition temperatures and enthalphy values before the expt.

purge gas flow rate has no effect on measurement of glass transition temperature, but effect on the measurement rate especially cooling rate. But it is better to keep the same rate during the whole set of samples

I believe that there's no effect on the measurement. Since you keep your equipment calibrated before all sets of experiments, there will not happen any  remarkable difference in the results.

If your dsc it's ok, the gas purge [if need only the dry electronic and fournace], don't interesting the transition.

Technically you don't change the Tg of the sample, but especially with gases like helium and argon you could see a difference in the rate of the transition.  This can effect the calculated Tg from the software.  But again I don't think you would see the change unless your DSC was really clean and properly calibrated.