This paper reports transient responses of graded longitudinal fins subject to step change in base temperature and base heat flux wherein the graded fin materials are theorized to have spatial- and temperature-dependent thermal conductivity. Microstructure variations in graded materials (GMs) are addressed by axially varying the thermal conductivity; because GMs are potentially high-temperature application materials, consequently, thermal conductivity and heat generation are, respectively, assumed as polynomial and linear functions of temperature. Additionally, most of the applicable pragmatic fluid regimes are accounted for using the power law convection coefficient. The numerical solution of a typical nonlinear governing differential equation is obtained by using a particle tracking-based method called the lattice Boltzmann method (LBM). The LBM is a mesoscopic-based simulation method centered around the principles of kinetic theory and statistical mechanics. The LBM formulation accompanied with the in-house MATLAB code of the aforesaid problem with varying parameters is reported; also, it is validated with a previously available solution. The foregoing analysis is carried out to enhance the performance of a fin by using the superior thermomechanical property of graded materials. Furthermore, the inclusion of temperature-dependent thermophysical properties and heat generation will provide more accurate design data. The reported graph reveals that, even though a linear GM fin tip possesses thermal conductivity that is 25% less in magnitude in comparison to the Type-II homogeneous material (HM-2), the GM fin always yields a higher fin tip temperature because of grading. In addition, the tip temperature deficits between GMs and HM-2 proportionally increase from 0.4 to 2.1% for values of $M$ increasing from 0.1 to 2.0, respectively, for step changes in temperature; whereas in the case of the step change base flux, the deficits increase from 8.72 to 12.1% for values of $M$ decreasing from 3.0 to 1.0, respectively.