Ash falling is a incident that happens right after the events in Blue Reflection Ray, during Blue Reflection Sun; and before Blue Reflection Tie/Second Light. The falling Ash is a consequence of the World System reset and the absence of an overseer. The previous overseer was Daath, who ended up defeated by Hinako at the end of Blue Reflection.

The Ash piled up in cities making them inhospitable, and also bringing up monsters that became threats that ultimately destroyed the world. There wasn't a direct source of Ash, it could appear anywhere, at any moment.

The Ash syndrome

Ash syndrome is a disease that happens as a direct consequence of Ash falling. People in contact with Ash become sick, and ultimately end up dying.

Timeline

Blue Reflection

Hinako defeats Daath, the World System deploys the Ash as a consequence of no overseer.
Overlaps in the Timeline mean that some events happen simultaneously between Blue Reflection and Blue Reflection Ray.

Blue Reflection Ray

The Ash starts to fall right after The Common was saved from Shino's attempts to control it. During the final episode, Mio can be seen holding Ash in her hand.

Blue Reflection Sun

The storyline of Sun is centered around the Ash and the Eroded. It is known from the Sun PV that the Ash is “a toxic substance that invades the body.”
More details to come

Blue Reflection Second Light

During Chapter 3, Shiho Kasuga comments on how her friends and herself fought against the enemies that appeared as a result of the Ash, which is confirmed by Kokoro Utsubo. Later it is known that Yuki died as a result of Ash Syndrome. Rena joined the AASA, looking for a cure.