Units that make it easier to think about a specific problem
You will have noted that the term s 2 / D t appears in the fundamental equations of diffusion. This term is dimensionless, as it must be, since it appears as an exponent, and exponents can't have a dimension. If you are going to solve a particular problem in diffusion, and want to give numerical results, perhaps as a graph, you want these results to be independent of the particular value of a diffusion constant. One way to do this is to plot the results for various values of s 2 / D t , and let the reader do the calculations for his/her particular case.
Another way to think about a diffusion problem is to adopt (just for that problem) special units of time or distance, so that the above term is 1. For example, suppose your original units were SI units (Kg, meter, second), and you were in love with the meter as a unit of distance. You could define a new unit of time, call it the Long, so that for 1 meter, s 2 / D t = 1. At a time of 1 Long, the material you were studying would change concentration by a characteristic amount at a distance of 1 meter. Since the diffusion constants of typical molecules is on the order of 10-10, the Long would be a long time indeed. However, if you were studying the diffusion of neutrons in a plasma with a temperature of a million degrees, maybe the time unit wouldn't be so long (I'm not familiar with hot plasmas, just a guess).
For molecules in water at room temperature it might be better to give up the meter, and switch to the micron. Then the unit of time would be more like a second, which is easier for most of us to comprehend. If the problem was in the field of biology, a micron might be better than a meter anyway, since cells have dimensions of the order of microns.