In Hemwick we encounter a few things but most importantly we encounter two types of enemies: the Executioners and the Witch of Hemwick. The Executioners appear to be enlarged, giant servants of the School of Mensis; the fact that they wear Yahar'gul Hunter attire and guard areas that Mensis has interest in support this connection. The Witch(es) of Hemwick are Eye Collectors, stronger versions of an enemy that is found in Yahar'gul. The fact that the Witch drops Bloodshot Eyeballs furthers this connection: "Material used in a Holy Chalice rite. An exquisite eyeball removed quickly after death, or perhaps even before."
We learn from items such as the Yellow Backbone, Bloodshot Eyeball, and Inflicted Organ that the Healing Church is using body parts as ritual items, and we find significant amounts of gravedigging, corpse-collecting, and body harvesting going on in Hemwick. We also find stagecoaches that leave from Hemwick, one of which brings us to Cainhurst.
Cut now to Yahar'gul, where we find stagecoaches lining the streets. After the Blood Moon has risen one of the stagecoaches will rumble and, if approached, a Cramped Casket will burst from it and attack the PC Hunter. This would seem to imply that caskets were being delivered en masse to Yahar'gul.
My interpretation of this evidence is that the denizens of Hemwick gathered corpses and body parts from both the living and the dead, packing them into caskets and shipping them to Yahar'gul. They were overseen by the Executioners and directed in their rituals by the Witch of Hemwick. It's also possible that many more people were sending their dead to Yahar'gul, most likely without knowing exactly what was going on. The Crowfeather Garb specifically refers to a "Blasphemous Yharnam burial service". One Executioner is found very early in the game amidst a massive collection of caskets in Central Yharnam, possibly gathering them to return to Yahar'gul.
It's very possible that the reason caskets line Yharnam is because they are collected by the School of Mensis for use in their experiments, with the citizens believing the Healing Church is gathering them for burial.