Middy instructions (https://github.com/middyjs/middy) are pretty sparse and unclear, through testing:

Observations

If throw within middlewares

the onError functions will be called in the same order as they are attached to Middy, not in reverse order like after functions are
if onError is an async function it will be awaited before processing passes to next middleware's onError
onError functions will be called for all middlewares, even those that have not yet passed begin function
because onError for all middlewares will be called, we cannot assume begin has already executed when processing onError function

do not call next(), it will in most cases result in remaining middleware onError's to be executed twice
return from async onError
can switch the Lambda to not throwing an error by any async onError function returning a falsy value
can stop remaining onError functions from being executed by returning a rejected promise, in an async onError function this might mean throwing an error (not tested)
if we want to Lambda to throw an error (or another middleware to choose whether to Lambda throws an error) we should return a truthy value

must end function using next(), next(handler.error), or Lambda's callback function
if return falsy seems to timeout, truthy (not sure) might throw error, but is not correct code
next() is the same as 'handling' the error, Lambda will not throw an error, all remaining onError's will execute, the Lambda response and error objects are null (nothing returned to calling resource)
next(handler.error) passes processing on to next onError, if all onError's pass on like this the Lambda will throw an error after all onError's executed
if an onError middleware invokes Lambda's callback we can set the response and error objects that the Lambda returns. remaining onError functions will be processed
the only was to return a response from the onError flow is to use the Lambda's callback