Auxins are a growth hormone produced in the meristems of plants.
Meristems are areas of plants where plant growth occurs. They are the site of repeated cell division of unspecialized cells. After division, these unspecialized cells differentiate and become specialized in relation to the function(s) they will perform. There are two types of meristems: lateral and apical.
Apical meristems are the site of primary growth in a plant, and are located at the root and shoot tips. Here you can find unspecialised cells, which follow this sequence to become a functional part of the plant:
-New unspecialised cells become available at the meristems (the site of mitosis)
-These cells become elongated and undergo vacuolation
-They become specialised (differentiated) to perform a particular function
-They then form part of a permanent tissue which performs a particular role within the plant
-The cycle continues for growth and regeneration purposes
Lateral meristems can be found growing laterally (where else?) to the plant. Lateral meristems are responsible for the thickening of the shoots and roots (structural support). This is why trees have rings....as they get older, they grow thicker. Every year a new "ring" forms around the core.
Needless to say, what we are interested in is the apical meristems, since this is where NEW root growth occurs. Now with that being said, lets get back to auxins....
Auxins are responsible in promoting cell elongation, a process that is required before differentiation of a cell. It is able to this by promoting the intake of water, increasing the elasticity of the cell to cope with the increase of water taken in by the cell. One of the most common auxins (and the one we are concerned with) is indole acetic acid (IAA).
Indole Acetic Acid affects the root and shoot tips of the plant as follows:
Shoot Tip - No matter what the concentration (of IAA), IAA promotes growth in the shoot areas of a plant. (Although higher concentrations promote growth more.)
Root Tip - High concentrations of IAA in the root tip inhibit growth, while small amounts are enough to promote growth.
Obviously we are interested in how IAA reacts in the root tip. Thus, clearly we do not want high concentrations of IAA in the roots. This would not promote faster growth.
Now, lets look at light's role in this debate. Believe it or not, light actually destorys the IAA auxins. But this only happens in substaintial amounts when there is alot of IAA to begin with. For instance, a clone (a shoot tip that has been cut), has alot of IAA at it's tip. And since the clone is so small, the amount of IAA in relation to it's entire mass is very high (if you have a 3" clone, and the top inch is the growing tip, it is filled with IAA, whereas in a full grown plant, there is much more 'mature' plant matter than growing tips). So in the case of the clone, light will destroy some of the auxins. And since the IAA auxins do not want to be destroyed, they will begin to "travel" downward. The light is what makes the IAA auxins travel to the bottom of a clone. Thus, no darkness is neede to make this happen.
And actually, it is what we want to happen. When we plant a clone, we want all of its energy to be directed into producing roots. By having the light on 24/7, we are destroying the auxins in the growing tip (while forcing them to move downwards), thus inhibiting plant growth, which allows all (or most) of the plants energy to be diverted to growing roots. And it is no coinsidense that this is where the IAA auxins happen to be headed. Since a newly cut clone has no roots to contain the IAA auxins already, the growing tip donates them in response to the constant bathing of light we give them.
So whether they know it or not, this is the real reason why many people put their clones under 24/7 lighting. If you were to give the clone a dark period, that would enable the auxins to begin to make the plant grow, thus using up valuable energy (and therefore time). However, since the roots are under soil or in a pot, having the light on 24/7 does not restrict root growth.
To sum things up, auxins are not only located or produced in the 'terminal bud', as you put it (which i'm not 100% what that is I assume you mean the shoot tips though). Auxins are located everywhere in a plant. But the ones that promote growth, the IAA auxins, are found (and produced) in the root tips AND shoot tips. So no 'travelling' of auxins is needed to promote faster root growth. After the donation of initial IAA auxins from the tip, the roots begin to manufacture their own IAA, thus allowing the shoot tip to keep its share of IAA, and begin to grow.
Whew, what a mouthful. I hope this helps more than hurts . Feel free to ask anymore questions...hopefully I (or someone) will be able to answer