This paper is concerned with the solutions of a two-component
generalisation of the Camassa-Holm equation. We examine the
propagation behaviour of compactly supported solutions, namely
whether solutions which are initially compactly supported will
retain this property throughout their time of evolution. In the
negative case, where we show that solutions have an infinite speed
of propagation, we present a description of how the solutions retain
weaker properties throughout their existence time, namely they decay
at an exponentially fast rate for the duration of their existence.