Temporal intensity fluctuation is one of the inherent features of fiber lasers. When utilizing the fiber lasers to pump a random Raman fiber laser (RRFL), the intensity fluctuation transfer from the pump to the random lasing could affect the output performance significantly. In this paper, we comprehensively compared the spectral, temporal, and power characteristics of an RRFL pumped by two different fiber lasers—a temporally unstable fiber oscillator and a temporally stable amplified spontaneous emission (ASE) source. Owing to less impact of the intensity fluctuation transfer, the ASE source-pumped RRFL shows ∼45.3% higher maximum output power, higher spectral purity (>99.9%) and optical signal-to-noise ratio (>40dB), weaker spectral broadening, and more stable temporal behavior compared to the fiber oscillator-pumped RRFL. Furthermore, based on the temporal-spatial-coupled Raman equations and the generalized nonlinear Schrödinger equations, we numerically revealed the impact of the pump intensity fluctuations on the output characteristics of RRFLs, and found that the temporal walk-off effect played an important role in the dynamics of intensity fluctuation transfer. This work may provide a reference for designing and implementing high-performance RRFLs and promote their practicability in sensing, telecommunications, and high-power applications.