Question

How does the rate of diffusion increase with increase in pressure?

Answer 1

Diffusion is proportional to the concentration gradient, which for gases is the same as the partial pressure gradient.
The rate is dependent on the velocity of the gas molecules, which depends on the temperature. The average velocity which should be used is the root mean square (i.e. the average based on the kinetic energy).
Pressure comes from kinetic energy of the molecules (hitting the sides of the container).
Thus the rate of diffusion is proportional to the square root of the pressure.
However note that at high pressures the diffusivity coefficient falls (difference between a real and ideal gas).