Context. Debris disks trace remnant reservoirs of leftover planetesimals in planetary systems. In the past years, a handful of "warm" debris disks have been discovered in which emission in excess starts in the mid-infrared. An interesting subset of these warm debris disks shows emission features in mid-infrared spectra, which points towards the presence of μm-sized dust grains, with temperatures above hundreds K. Given the ages of the host stars, the presence of these small grains is puzzling, and raises questions about their origin and survival in time.Aims. This study focuses on determining the mineralogy of the dust around seven debris disks with evidence for warm dust, based on Spitzer/IRS spectroscopic data, to provide new insights into the origin of the dust grains. Methods. We developed and present a new radiative transfer code (Debra) dedicated to spectral energy distribution (SED) modeling of optically thin disks. The Debra code is designed such that it can simultaneously determine dust composition and disk properties. We used this code on the SEDs of seven warm debris disks, in combination with recent laboratory experiments on dust optical properties. Results. We find that most, if not all, debris disks in our sample are experiencing a transient phase, suggesting a production of small dust grains on relatively short timescales. Dust replenishment should be efficient on timescales of months for at least three sources. From a mineralogical point of view, we find that crystalline pyroxene grains (enstatite) have low abundances compared to crystalline olivine grains. The main result of our study is that we find evidence for Fe-rich crystalline olivine grains (Fe/[Mg + Fe] ∼ 0.2) for several debris disks. This finding contrasts with studies of gas-rich protoplanetary disks, where Fe-bearing crystalline grains are usually not observed. Conclusions. These Fe-rich olivine grains, and the overall differences between the mineralogy of dust in Class II disks compared to debris disks suggests that the transient crystalline dust in warm debris disk is of a new generation. We discuss possible crystallization routes to explain our results, and also comment on the mechanisms that may be responsible for the production of small dust grains.