This paper investigates inertialess channel flow of elastic liquids having continuously stratified constitutive properties. We find that an Oldroyd-B fluid having a sufficiently rapid normal stress variation shows instability. The mechanism is the same as for the two-fluid co-extrusion instability that arises when elasticity varies discontinuously. We find, using numerical and asymptotic methods, that this mechanism is opposed by convective effects, so that as the scale over which the elastic properties vary is increased, the growth rate is reduced, and finally disappears. A physical explanation for the stabilisation is given. Regarding an Oldroyd-B fluid as a suspension of Hookean dumbbells, we show that a sufficiently steep variation in dumbbell concentration (with attendant rapid changes in both viscosity and elasticity) will provide an instability of the same kind. Finally we show that Lagrangian convection of material properties (either polymer concentration or relaxation time) is crucial to the instability mechanism. A White-Metzner fluid having identical velocity and stress profiles in a channel flow is found to be stable. The implications for extrudate distortion, and constitutive modelling are briefly discussed.