The radial structure of intra-cavity light pulses is considered in the hole-coupled free-electron laser (FEL) oscillators of the Dutch FEL facility, FELIX. In a previous numerical study, it was shown that the radial profile can vary significantly over the light pulse, in particular when the electron pulses are short with respect to the slippage length. In this paper experimental indications are presented for the predicted transverse profile of the radiation. The decay of the power coupled out through the hole has been measured, after the electron beam was switched off, for the two different resonators employed in FELIX. These resonators differ in cavity length, mirror curvature and aperture radius, and in the width of the vacuum tube inside the undulator. The decay rate of the extracted radiation is interpreted according to the calculated intra-cavity mode structure from ELIXER. This simulation code describes the time-dependent radial profile in terms of a finite number of Gauss-Laguerre functions. Two extreme cases are discussed in further detail, one in which edge losses are large and one in which they are small.