At this time the only thing that is eligible to run locally is one and only one smart app: smart lighting. And then only for a few specific device types. For example, any lighting rule with Phillips hue in it will not run locally.
They have said that there are plans to expand local processing, but for now it is what it is.
As for the recent platform issues, see the following:
The FAQ in the community created wiki has links so you can see what is running locally at your own house: