standard model - How (or when) do gluons change the color of a quark?

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?

Answer

The idea that baryons contain three quarks is a significant oversimplification wrong. 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.

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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.