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The investigation of wave propagation and instabilities in plasmas requires the knowledge of the constitutive relation, i.e. the relation between oscillating wave electric field and current in the plasma. The constitutive relation in a hot nonuniform plasma has an integral non-local form: the current at a given point depends on the fields at other points. Explicit expressions for the constitutive relation were previously obtained only for very special cases: restrictive approximations were mostly introduced at the early stages of derivation to simplify the form of the constitutive relation. In the present work, the constitutive relation of a hot magnetised plasma is derived directly from the linearised Vlasov equation for the distribution function of plasma particles without making any assumption other than the validity of the drift approximation for the description of the particle orbits in the static magnetic configuration. In the integrals which define the oscillating plasma current, a change of integration variables from the position of particles to the position of the guiding centres of particles has allowed us to apply mathematical techniques similar to those of the uniform plasma limit to perform the expansion in harmonics of the particle cyclotron frequency. The constitutive relation is written in integral form as a convolution of Fourier components in each direction of plasma inhomogeneity. Since the general Fourier representation for the wave electromagnetic field is used, the wave equations obtained are valid in a wide range of frequencies and wavelengths. The general constitutive relation has been specialised to obtain the wave equations describing low frequency drift and shear Alfven waves, which play an important role in tokamak plasma stability, providing a mechanism for the generation of plasma microturbulence. These wave equations generalise those of the gyro-kinetic theory, based on a simpler gyro-kinetic equation derived by averaging of the Vlasov equation on the timescale of the fast particle gyro-motion. Exploiting the fact that these waves propagate mostly in the diamagnetic direction (the direction perpendicular to the directions of the equilibrium magnetic field and to its gradient, the integro-differential equations have been simplified and put into a form which is essentially equivalent to the wave equations of the gyro-kinetic theory. Namely, the equations obtained are differential in the radial variable and take into account the finite Larmor radius effects to all orders along the diamagnetic direction, since the wavelengths can be of the order of the thermal ion Larmor radius. The wave equations obtained in this way are in a form suitable for numerical solution with standard methods, for example with finite elements in the radial variable, and thus offer a good starting point for applications.