We present a multichannel finite-range three-body scattering theoretical model used for investigating the magnetically tuned atom-dimer collision reaction. We take the $^{6}\mathrm{Li}\text{\ensuremath{-}}^{6}\mathrm{Li}_{2}$ system as an example to calculate the atom-dimer scattering length and dimer relaxation rate steered by a magnetic field. When the reactants are prepared in three shallow atom-dimer hyperfine channels, our calculation results agree with two experimental measurements [S. Nakajima et al., Phys. Rev. Lett. 105, 023201 (2010); T. Lompe et al., Phys. Rev. Lett. 105, 103201 (2010)]. We find that the two-body $p$-wave interaction has a significant influence on the atom-dimer collision reaction in the three hyperfine channels. By including the hyperfine and Zeeman interactions, we calculate the hyperfine state-to-state transition probability from the reactant channel to the deep atom-dimer hyperfine channel. We find a universality in the atom-dimer collision reaction, i.e., the universality of deep dimer product. The universality of deep dimer product in the $^{6}\mathrm{Li}\text{\ensuremath{-}}^{6}\mathrm{Li}_{2}$ collision reaction is more robust at magnetic field $B>620$ G than that at $B<620$ G.