I know a baryon is only stable when it contains a quark of each color. And as far as I know, the gluon essentially changes the color of a quark and moves onto the next, and this is what holds the particles together. But in the process of the gluon moving from one quark to the next, wouldn't the baryon have two quarks of the same color, making it unstable? Or does the gluon move instantaneously, or is the baryon not unstable enough to decay before the gluon reaches the next quark? Or… essentially, how does this process actually work?
[Physics] How (or when) do gluons change the color of a quark
color-chargegluonsquantum-chromodynamicsquarksstandard-model
Related Solutions
It boils down to whether one can make a color neutral particle with a quark and some colored gluons.
The rationale for the concept of color can be highlighted with the case of the omega-minus, a baryon composed of three strange quarks. Since quarks are fermions with spin 1/2, they must obey the Pauli exclusion principle and cannot exist in identical states. So with three strange quarks, the property which distinguishes them must be capable of at least three distinct values.
See also the evidence for three colors.
Now the gluons do not carry a single color
Gluon interactions are often represented by a Feynman diagram. Note that the gluon generates a color change for the quarks. The gluons are in fact considered to be bi-colored, carrying a unit of color and a unit of anti-color as suggested in the diagram
I do not think that the algebra of the existing model can model what you want. If it is a red quark it might be neutralized by a gluon carrying antired, but there would be left over the other color carried by the gluon. This would be true for combinations of gluons too, there would be left over color or anticolor, so color neutrality cannot be attained within QCD.
Flavor is entirely orthogonal to color - the gluon(s) neither "know" nor "care" about flavor, but the existence of different quarks still leads to phenomena descending from the strong force that you wouldn't get without them:
What remains of the strong force on the scale between nucleons is often called the residual strong force and can be thought of as being mediated by pions. The pions, being bound states of up- and down-quarks, evidently would not exist in a world with only a single quark.
Another phenomenon related to the strong force is that confinement depends on the number of flavors, too, see e.g. this answer.
So while on the level of the QCD Lagrangian there is nothing that would indicate at first sight that the different quark species and gluons interact in interesting ways (and indeed a theory with only a single quark species and gluons would be consistent!), there are emergent phenomena that depend on both gluons and different flavors.
Best Answer
The idea that baryons contain three quarks is
a significant oversimplificationwrong. It works for some purposes, but in this case it causes way more confusion than it's worth. So you should stop thinking of baryons as groups of three quarks and start thinking of them as excitations in quantum fields - and in particular, excitations in all the quantum fields at once. Quark fields, gluon fields, photon fields, and everything. These excitations propagate through spacetime and convert among each other as they go, and in a baryon, the propagation and mutual conversion happen to sustain each other so that the baryon can exist as a coherent particle for a while.One of the conditions required of all these excitations in fields is that they be a color singlet, which is the strong interaction's version of being uncharged. There's a simple intuitive justification for this: just as an electrically charged particle will tend to attract oppositely charged particles to form neutral composites (like protons and electrons attracting each other to form atoms), something which has the charge associated with the strong interaction (color charge) will attract other color-charged particles to form neutral composites (color singlets).
Now, if you literally only had three quarks, the only way to make them a color singlet is to have one be red, one be green, and one be blue.1 (Or the anticolor equivalents.) But with all the complicated excitations that make up a baryon, there are all sorts of ways to make a color singlet. You could have three red quarks, a green-antired gluon, and a blue-antired gluon. Or two red quarks, two green quarks, an antiblue antiquark, a blue-antired gluon, and a blue-antigreen gluon. Or so on; the possibilities are literally infinite.
The point is that you don't actually have to have a quark of each color in the baryon at all times. Only the total color charge in the baryon matters.
Given that, it should seem reasonable that gluons change the color of quarks whenever they are emitted or absorbed, in a way that keeps the total color charge the same. For example, a blue quark could absorb a green-antiblue gluon and become a green quark.
1I'm glossing over some quantum-mechanical details here; specifically, a color singlet wavefunction needs to be an antisymmetrized linear combination, like $\frac{1}{\sqrt{6}}(rgb - rbg + gbr - grb + brg - bgr)$, not just $rgb$. But as long as you don't worry about which quark is which color, for purposes of this answer it's safe to ignore this.