  We define a Fourier-Mukai transform for a triple consisting of two
holomorphic vector bundles over an elliptic curve and a homomorphism between
them. We prove that in some cases the transform preserves the natural stability
condition for a triple. We also define a Nahm transform for solutions to
natural gauge-theoretic equations on a triple -- vortices -- and explore some
of its basic properties. Our approach combines direct methods with dimensional
reduction techniques, relating triples over a curve with vector bundles over
the product of the curve with the complex projective line.
