Propagation is a standard way of producing certain new conformal blocks from old ones that corresponds to the geometric procedure of adding new distinct points to a pointed compact Riemann surface. On the other hand, sewing conformal blocks corresponds to sewing compact Riemann surfaces. In this article, we clarify the relationships between these two procedures. Most importantly, we show that, "sewing and propagation are commuting procedures".

The proof of this result relies on establishing the propagation of conformal blocks associated to holomorphic families of compact Riemann surfaces. We prove this in our paper using the idea that, "propagation is itself a sewing followed by an analytic continuation". This result generalizes previous ones on single Riemann surfaces [Zhu94,FB04], and supplements those on algebraic families of complex algebraic curves [Cod19,DGT19a].

The author was supported by BMSTC and ACZSP (Grant no. Z221100002722017).