Dispersion of a Gaussian wave packet- why does momentum become more certain?

So I understand from the equations, and considering the Fourier components of a Gaussian, how the wavefunction for a free particle spreads due to dispersion. However mathematically, and from the reciprocity relation, this means the momentum is becoming more certain! Is there a physical process governing this? Or is there a way to understand why this occursvwithoit simply referring to the position wavefunction and its spreading?

If you measure the position of the particle to be at $x=x_0$ at the instant $t=0$, then, you can measure it quickly again and it will have to be close to $x_0$, but you don't know very well where it is going. That is a distribution narrow in position and broad in momentum.