The knowledge of lateral heterogeneities is crucial for path corrections in moment tensor inversions using surface waves. After some attempts to use regionalized Earth models for very long-period surface-wave moment-tensor inversions, recent tomographic Earth models offer the possibility to make short-period path corrections and therefore retrieve more reliable moment tensors for teleseismic earthquakes. First we try to evaluate the precision required for path corrections in comparison with source effects. Some selected Earth models are tested to evaluate how their results compare to those using multiple-frequency filtering techniques. Some real cases illustrate the sensitivity of moment-tensor solutions to the different path corrections, and it appears clearly that regionalized Earth models and tomographic models deduced from long-period data alone (greater than 150 sec) cannot lead to trustworthy broadband moment-tensor inversions. Recent tomographic models using phase velocities at much shorter periods (40 to 200 sec) offer a precision comparable to that of the multiple-frequency filtering technique. Both methods lead to acceptable source mechanisms, using a small number of stations, in more than two cases out of three. The use of recent global tomographic models based upon shorter-period surface waves might thus be a useful alternative to heavy multiple-frequency filtering techniques to automate source studies, especially for rapid determinations using a small number of stations.