Heterochromatin can be epigenetically inherited in cis, leading to stable gene silencing. However, the mechanisms underlying heterochromatin inheritance remain unclear. Here, we identify Fft3, a fission yeast homolog of the mammalian SMARCAD1 SNF2 chromatin remodeler, as a factor uniquely required for heterochromatin inheritance, rather than for de novo assembly. Importantly, we find that Fft3 suppresses turnover of histones at heterochromatic loci to facilitate epigenetic transmission of heterochromatin in cycling cells. Moreover, Fft3 also precludes nucleosome turnover at several euchromatic loci to prevent R-loop formation, ensuring proper replication progression. Our analyses show that overexpression of Clr4/Suv39h, which is also required for efficient replication through these loci, suppresses phenotypes associated with the loss of Fft3. This work uncovers a conserved factor critical for epigenetic inheritance of heterochromatin and describes a mechanism in which suppression of nucleosome turnover prevents formation of structural barriers that impede replication at fragile regions in the genome.