From the old NSW DPI document about growing finger limes:
'The main disease affecting finger limes is melanose (Diaporthe citri), a fungal disease that causes dark brown to black spots on the foliage, twigs and fruit. Spores of the fungus develop in dead citrus tissue and are released by water and/or rainfall. The fungus affects all citrus varieties and the incidence of melanose usually increases as trees age and the amount of dead wood in the canopy increases. Annual light pruning should aim to remove any dead wood. Experienced finger lime growers recommend not applying any chemical sprays whilst the trees are flowering as the flowers may be damaged.
Finger limes also occasionally suffer twig or branch dieback, but no causal organism has yet been identified. Dieback in other citrus varieties can be caused by a range of factors including frost injury, hot dry or very cold winds, or some other factor resulting in the plant being unable to get sufficient water when needed, such as a lack of soil moisture or damage to the root system. In coastal orchards the melanose fungus can exacerbate twig dieback.'
Growers have suggested everything from heavy soils to over fertilisation as being causes for dieback. windrub is another factor as overcrowding of branches can cause windrub and the branches throw leaves under stress and die back. Even very healthy trees can carry quite a bit of die back and in the wild they carry lots of dead wood due to the very dense nature of the tree. FLs naturally grow in very rich ground sometimes in creek beds. So they handle wet conditions well on their own roots and some wild finger limes have been recorded as the most resistant of all citrus to citrophora (citrus dieback).