Many people say that when the transfer the betta fish (beta fish) to a new bowl or adding new water to the old bowl or tank itself, you should not use complete new water but slowly add new water to the bowl little by little. I do not believe in this. The point is you just need to maintain the pH and temperature of the new water. The pH should be slightly acidic and the water temperature around 80 degrees F. The main reason we change in water of the betta fish/beta fish bowel is the water will be contaminated with the feces.
Before transferring the betta fish to the new water. You just check the pH and temperature of the new bowl know that the temperature and pH and of optimum range and transfer the betta fish to the new bowl or tank. If you want to transfer the betta fish back to the its original bowl, just clean the bowl and transfer if along with the new water.
Your betta fish care of the beta fish depends on you being a little bit careful while feeding and changing the water.