We study the Gibbsian character of time-evolved planar rotor systems (that is, systems which have two-component, classical XY, spins) on Z^d, d=2, in the transient regime, evolving with stochastic dynamics and starting from an initial Gibbs measure ¿. We model the system with interacting Brownian diffusions X=(Xi(t)), t>=0; i e Z^d moving on circles. We prove that for small times t and arbitrary initial Gibbs measures ¿, or for long times and both high- or infinite-temperature initial measure and dynamics, the evolved measure ¿^t stays Gibbsian. Furthermore, we show that for a low-temperature initial measure ¿ evolving under infinite-temperature dynamics there is a time interval (t0,t1) such that ¿^t fails to be Gibbsian for d=2. Keywords: Gibbs property; Non-Gibbsianness; Stochastic dynamics; XY-spins.