How do color values change mathematically as you change temperature and tint?

Adobe actually describes extremely well their white balancing process in the Digital Negative (DNG) Specification, specifically in Chapter 6 Mapping Camera Color Space to CIE XYZ Space.

The process is not super trivial but the core principle is to compute a matrix converting from Camera Space, i.e. space the captured image is upon capture, to CIE XYZ tristimulus values under D50 illuminant for a given set of white balance chromaticity coordinates xy.

With the chromaticity coordinates, it is possible to compute the corresponding correlated colour temperature (CCT), i.e. colour temperature and D_uv, using Robertson (1968) method.

Once the CCT is computed, the matrix from Camera Space to CIE XYZ is obtained via linear interpolation of the ColorMatrix1 and ColorMatrix2 matrices that Adobe (or other vendors following the DNG specification) provide for the camera, and typically embedded in a DCP profile, or EXIF/XMP metadata. The two matrices are commonly given for tungsten-like light source, e.g. Illuminant A (CCT ~= 2856K) and daylight-like, e.g. D65 (CCT ~=6505K) and transform from CIE XYZ to camera space.

There are more details in the specification, but this is really the core of the algorithm.

We have a full Python implementation in colour-hdri in the colour_hdri.models.dng module. An example would be like that:

M_color_matrix_1 = np.array(
    [[0.5309, -0.0229, -0.0336],
     [-0.6241, 1.3265, 0.3337],
     [-0.0817, 0.1215, 0.6664]])
M_color_matrix_2 = np.array(
    [[0.4716, 0.0603, -0.0830],
     [-0.7798, 1.5474, 0.2480],
     [-0.1496, 0.1937, 0.6651]])
M_camera_calibration_1 = np.identity(3)
M_camera_calibration_2 = np.identity(3)
analog_balance = np.ones(3)
XYZ_to_camera_space_matrix(  # doctest: +ELLIPSIS
    np.array([0.34510414, 0.35162252]),
    2850,
    6500,
    M_color_matrix_1,
    M_color_matrix_2,
    M_camera_calibration_1,
    M_camera_calibration_2,
    analog_balance)

array([[ 0.4854908...,  0.0408106..., -0.0714282...],
        [-0.7433278...,  1.4956549...,  0.2680749...],
        [-0.1336946...,  0.1767874...,  0.6654045...]])

After some more research I think I have an answer to my question. I'm saying the other answers are wrong, they sound very informed, but this is just what I've found about how the sliders of color temperature work, though not necessarily how cameras work or auto-WB or LR/ACR's algorithm for interpreting metadata.

I watched a YouTube video about how to adjust the temperature and tint in Photoshop without going into ACR and what the instructor recommended was to go into LAB mode and open up the curves and in the A and B channel, scoot the top and bottom node in until they are touching the spike in the histogram. If you want to add more or less temperature or tint, adjust one of the nodes so that it is asymmetric, so that it is a little more to the left or right of the center than the other. I would say this method is more finicky than LR or ACR, as anyone whose messed with the LAB curves could guess, but it just goes to show that that's probably what's going on under the hood.

I deduced from this that while XYZ and LAB and matrixes must all be in play when picking colors, the LAB slider's job is only to create the opposite color of the color cast in order to cancel it out, and that can be done with simple LAB mixing.

LAB primaries are a good choice not only because the LAB color space is big enough to fit all the other color spaces, but also because it makes white balancing really simple for beginners. Anything on the warm end of the Kelvin scale can be canceled out with blue and green, which both slide to the left. Colors on the cool end of the Kelvin scale can be canceled out with yellow and magenta, which are both to the right. Nice and easy for the end user.