Forget about Feynman lectures, what you need here is the convection-diffusion PDE:
$$\frac{\partial c}{\partial t}=\nabla(D\nabla c)-\nabla(\mathbf{v}c)$$
This is essentially a version of the transport equation.
Assuming $D$ and $\mathbf{v}$ to be constant and for the $text{1D}$$\text{1D}$ case we have:
$$c_t=Dc_{xx}+v c_x$$