What generally happens to the temperature of red when a little blue is mixed into it?

Correct answer: It becomes a relatively cool red

The governing concept is directional colour bias. Adding a little blue shifts red toward violet or bluish red. Since blue and violet are on the cool side of the colour spectrum, the resulting red is relatively cooler than the original red. It does not automatically become neutral, and it does not become warmer; those outcomes would require a different mixture or a failure to recognise the hue shift.