A key issue with small quantities of ash, ESPECIALLY if there are repeated ash encounters, is presumably going to be build-up of accumulations in confined spaces, such as inside blades, in the air galleries serving the cooling holes. Unless blade cooling fails, this ash presumably would NOT melt but seems to me there must be plenty of possibilities for lumps of it to build up. Of course, if the air-flow IS impeded, then the blade overheats, the ash melts and the blockage becomes permanent and potentially very damaging.
What I've not seen yet is any comment about possibilities of removing ash post-flight BEFORE it builds up and melts.... Engine washing (eg. P&W Eco Power) is already in use for other reasons - maybe it would also be relevant for ash.
This is a suitable discussion for the Tech section
The only time that the ash is going to melt is when it's entrained in the compressor air as it passes through the stoichrometric heat of the burner. Once it reaches the cooler turbine it's either melted or not. Once you are out of the cloud the residual ash is expected to be only lodged in crevices where it can't get blown away.
How long it stays in those crevices is up to the operator and their maintenance procedures (detect and remove)