I’ve seen this rule for pumped milk: never add a new portion of warm milk to the portion pumped earlier and kept in the fridge, until the new milk cools down to the same temperature. Adding warm to warm is also fine. Why?
Warmer milk is a much better place for bacteria to grow than is cold, however they do still grow in cold.
Adding warm milk to cold then makes a brew good for bacterial growth that due to the older age of the cold milk already had more bacteria in it to start with.
Warm milk tends to be low on bacteria. Its either fresh or the process of heating it may have killed some already.
The temperate GREATLY affects the rate at which chemical reactions happen, including the reactions that keep bacteria alive and allow them to soil foods. When adding warm to cold, you are raising the temperature of the stuff rhat was cold, which may include potential pathogens/organisms that will spoil the milk. By allowing the new milk to cool you remove the risk of activating anything that was already present, but dormant because of the cold.