  Asymptotic homogenisation is considered for problems with integral
constraints imposed on a slowly-varying microstructure; an insulator with an
array of perfectly dielectric inclusions of slowly varying size serves as a
paradigm. Although it is well-known how to handle each of these effects
(integral constraints, slowly-varying microstructure) independently within
multiple scales analysis, additional care is needed when they are combined.
Using the flux transport theorem, the multiple scales form of an integral
constraint on a slowly varying domain is identified. The proposed form is
applied to obtain a homogenised model for the electric potential in a
dielectric composite, where the microstructure slowly varies and the integral
constraint arises due to a statement of charge conservation. A comparison with
multiple scales analysis of the problem with established approaches provides
validation that the proposed form results in the correct homogenised model.
