We present a behavioural typing system for a higher-order timed calculus, using types to model timed protocols (i.e., application-level specifications of interaction patterns among distributed applications), and the calculus to abstract implementations. Behavioural typing ensures that processes in the calculus will perform actions in the time-windows prescribed by their protocols. We introduce duality and subtyping relations for timed asynchronous session types. Duality includes a class of protocols that previous work on asynchronous timed session types could not type-check. Subtyping is critical for precision of our typing system, especially in the case of session delegation. The composition of dual timed asynchronous types enjoys progress when using an urgent receive semantics, in which receive actions are executed as soon as the expected message is available. Our calculus increases the modelling power of calculi used in the previous timed session work. Specifically, we add a blocking receive primitive with timeout, and a primitive that consumes an arbitrary amount of time in a given range.