In the present study, (Fe41Co7Cr15Mo14Y2C15B6)100−xCux (x = 0, 0.25 and 0.5 at.%) amorphous alloys were prepared by copper-mold casting. To clarify the effect of the minor addition of copper on the mechanism of nucleation and growth during the crystallization process, an isokinetic analysis was performed. The activation energies (E) of the various crystallization stages were calculated by using theoretical models including Kissinger–Akahira–Sunose (KAS), Flynn–Wall–Ozawa (FWO), Augis–Bennett and Gao–Wang methods. In addition, Augis–Bennett, Gao–Wang and Matusita methods were used to investigate the nucleation and growth mechanisms and to determine other kinetic parameters including Avrami exponent (n), the rate constant (Kp) and dimensionality of growth (m). The obtained results revealed that the activation energy—as well as thermal stability—was changed with minor addition of copper. In addition, the obtained Avrami exponent values were confirmed by Johnson–Mehl–Avrami–Kolmogorov (JMAK) method. The research findings demonstrated that the value of Avrami exponent is changed with minor addition of copper, so that the Avrami exponents of all crystallization stages, except the second peak for copper-free amorphous alloy, were equal to integer values ranging from two to four, indicating that the growth mechanisms were controlled by interface. Moreover, the kinetic parameters of n and b for all peaks were increased by an increase in crystallization temperature, which can be attributed to the increase in the nucleation rate.