1. A system comprising: a computer-readable memory storing executable instructions; and one or more processors in communication with the computer-readable memory, wherein the one or more processors are programmed by the executable instructions to at least: receive audio data from a client computing device separate from the system, wherein the audio data comprises data regarding an utterance of a user; produce first speech processing results using a base speech processing model and the audio data, wherein the base speech processing model is stored at the system; obtain a specialized speech processing model from a network-accessible data store separate from the system and separate from the client computing device, wherein the obtaining is initiated by the system subsequent to receipt of the audio data and prior to completion of producing the first speech processing results; determine, based at least partly on a time at which the specialized speech processing model is obtained, that the system is to produce second speech processing results using the specialized speech processing model subsequent to initiating production of the first speech processing results; and produce the second speech processing results using the specialized speech processing model and at least one of the audio data or the first speech processing results.