In this paper, we define a compound generalized fractional counting process (CGFCP) which is a generalization of the compound versions of several well-known fractional counting processes. We obtain its mean, variance, and the fractional differential equation governing the probability law. Motivated by Kumar et al. (2020), we introduce a fractional risk process by considering CGFCP as the surplus process and call it generalized fractional risk process (GFRP). We study the martingale property of the GFRP and show that GFRP and the associated increment process exhibit the long-range dependence (LRD) and the short-range dependence (SRD) property, respectively. We also define an alternative to GFRP, namely AGFRP which is premium wise different from the GFRP. Finally, the asymptotic structure of the ruin probability for the AGFRP is established in case of light-tailed and heavy-tailed claim sizes.