Interesting question, I am not sure if I have a definite answer, but at least some ideas:
The pigment in the hair is made by specialized cells, the melanocytes. The make the pigment (eumelanin=dark and pheomelanin=red/yellowish) which is then deposited into the growing hair. They are located at the bottom of hair bulb and usually die at the end of each hair cycle. To have the hair of the next hair cycle colored as well, this melanocyte population in the hair bulb needs to be replenished. This is done by melanocate stem cells, which live in the hair bulge and which start to reproduce and differentiate in melanocyte which then migrate down to the hair bulb. There is an excellent article following this process:
The hair later gets grey (low pigment) and white (no pigment) when these melanocytes are not, or not fulley replenished in later hair cycles. Why this happens, there are a few hypothesis.
First, its possible, that with age the melanocyte stem cell population (which not only need to replenish the melanocytes, but also need to maintain its own population) is getting smaller and disappears over time. This process would explain gradually graying over time. See this reference:
Another possibility is the presence of reactive oxigen species (ROS) which occur naturally in melanocytes. Over time, the enzyme which intercepts this ROS (catalase) is getting less active or present in the cells and thus oxidative damage is getting bigger, damaging the melanocytes and their ability to make pigment. It would actually bleach itself...
Melanin has also an important role in intercepting ROS, so in this process less and less pigment is made (as the damage accumulates) and this also leads to hair, which is less pigmented for exogeneous oxidative stress:
It could be possible (and this is a wild guess) that the phenomenon of the tips graying first is situated by this. Hair less protected against exogeneous oxidative stress (smoke, UV etc.) so the melanin is "used up" during this until the hair greys at the tip, since its the oldest and most exposed part of the hair. This article by Tobin gives also a nice overview: