Thin flat dielectric films can be low-macroscopic-field (LMF) electron emitters, able to generate electrons when subject to a macroscopic electric field in the range 1-50 V mum(-1). This phenomenon is a known cause of pre-breakdown currents in high-voltage vacuum breakdown, and is now the basis of a broad-area electron-source technology, using carbon-based thin films and other materials. The phenomenon occurs because the dielectric film is, or becomes, an electrically nanostructured heterogeneous (ENH) material, with quasi-filamentary conducting channels between its surfaces. These channels connect to emitting features near or on the film/vacuum surface, or act as electron emitters themselves. The film may contain conducting or semiconducting particles that assist with conductivity and/or act as emitting features. Several forms of thin-film LMF emitter exist: in each case the situation geometry ensures that sufficient field enhancement occurs at the 'tip' of the emitting feature for the emission process to be some form of tunnelling field electron emission (probably 'cold' in some cases, 'hot' in others). This paper explores aspects of the theory of thin-film LMF emission, starting from the need to understand the behaviour of emitters based on amorphous carbon films. A summary review, with extensive references, is given of relevant past work outside the immediate 'carbon field emission' context. Relevant aspects of semiconductor field emission theory are noted. Comment is made on the original experiments on diamond field emission, and on theoretical misconceptions in the carbon field emission literature. Analysis of carbon-film emitter behaviour suggests that emission must primarily be due to geometrical field enhancement, that in at least some cases arises from conducting nanostructure inside the film. In one case, published film characteristics can be used to show that sufficient field enhancement should be available. Some problems with an 'internal field enhancement' hypothesis are considered and disposed of. Difficulties with Latham's theory of field-induced emission from ENH materials are pointed out: a new theory, largely qualitative at this stage, can explain longstanding problems: this assumes that dielectric films must be treated as 'hopping conductors' not semiconductors. Electron emission takes place via localised surface states: transition to a channel-limited current regime takes place when the surface states no longer have high enough occupation probability to screen the external field, and is accompanied by anomalous band bending at the channel tip, Mathematical theories of band bending and field emission for hopping conductors are required. Some consequences for the design of LMF emitters are noted.