In this paper, we analyze the performance of a reconfigurable intelligent surface (RIS)-assisted multi-hop transmission by employing multiple RIS units to enable favorable communication for a mixed free-space optical (FSO) and radio-frequency (RF) system. We consider a single-element RIS since it is hard to realize phase compensation for multiple-element RIS in the multi-hop scenario. We develop statistical results for the product of the signal-to-noise ratio (SNR) of the cascaded multiple RIS-equipped wireless communication. We use decode-and-forward (DF) and fixed-gain (FG) relaying protocols to mix multi-RIS transmissions over RF and FSO technologies and derive probability density and distribution functions for both the relaying schemes by considering independent and nonidentical double generalized gamma (dGG) distribution models for RF transmissions with line-of-sight (LOS) and inverse-Gamma shadowing effect and atmospheric turbulence for FSO system combined with pointing errors. We analyze the outage probability, and average bit-error rate (BER) performance of the considered system. We also present an asymptotic analysis of the outage probability using gamma functions to provide insight into the considered system in the high SNR regime. We use computer simulations to validate the derived analytical expressions and demonstrate the performance for different system parameters on the RIS-assisted multi-hop transmissions for a vehicular communication system.