Paint Texture Tool doesn't respect Brush

The Paint Texture Tool doesn't consider the brush while dragging / painting. Example:



Left is a single click (green) with the brush, looks okay. Right is the same brush being used to paint the texture, dragging the brush from top to bottom. I would have expected the fading of the stroke to have a falloff the same way the brush has. And the texture itself is painted even outside of the brush itself.

Any ideas how to prevent this problem and keep the brush painting the texture the way it's supposed to be? It shouldn't bleed out imo.