There is no evidence that cold drinks during or after meals adversely affects health - in fact - the opposite may be true.
In studies of people with pure liquid meals, the temperature of the liquid had essentially no effect on the time it took for the stomach to empty itself (i.e. digest the food). Additionally, it is worth noting that a liquid as it travels down the esophagus is warmed and becomes almost equivalent to body temperatures within minutes. Whatever effect it would have would be gone quickly.
It is worth noting that the body heats up while digesting food (it is a metabolically active process.) That actually suggests that cool drinks actually make it metabolically easier for the body to digest food because it doesn't have to work as hard to cool down after eating. That also suggests that you can burn a few more calories if you have a hot drink after eating because your body will have to work to cool the blood and maintain your temperature (i.e. homeostasis)