I have cultured hela cell line in DMEM medium for many time and didn't see any affection on their growth or other parameters but if you want to change the culture medium, you have to do this step by step.
First use the EMEM 90% and DMEM for remained 10% and then increase the DMEM percentage with a constant rate up to 100%.
But be careful that the defreezing process and first culture have to be done by its own medium.
EMEM contains minimum concentrations of nutrients required for cell growth in culture while DMEM is a much more complex media with increased concentration of media components. Moreover, DMEM includes more substitutes that increase the nutrient condition of the media.
So, there is no harm in shifting from EMEM to DMEM. But as Saadeh mentioned change of media must be done gradually starting from 90% EMEM + 10% DMEM and then gradually increasing the percentage of DMEM and decreasing the percentage of EMEM till you reach 100% DMEM. This will take some time for cells to get adapted to the new medium, DMEM.