Fluorescence line narrowing measurements have been carried out between the lowest Stark components of the ${}^{2}{\stackrel{\ensuremath{\rightarrow}}{{F}_{5/2}}}^{2}{F}_{7/2}$ transition of ${\mathrm{Yb}}^{3+}$ ions in fluoroindate glasses. At low concentration (0.15 mol % of ${\mathrm{Yb}}^{3+}$) the luminescence spectrum consists in a narrow band which repeats the exciting profile, so the migration processes between ${\mathrm{Yb}}^{3+}$ ions are not important at this concentration. However, at higher concentrations (0.75 or 2.25 mol %) the luminescence spectrum contains moreover a broad band produced by other ions nonexcited initially (background fluorescence). A model is proposed to describe the temporal evolution of the narrow band fluorescence and the shape of the background fluorescence. A very good agreement with the experimental results is found taking into account a dipole-quadrupole $(S=8)$ nonradiative energy transfer process assisted by a phonon.