This study focuses on the protection of soft-biometric attributes related to the demographic information of individuals that can be extracted from compact representations of face images, called embeddings. We consider a state-ofthe-art technology for soft-biometric privacy enhancement, Incremental Variable Elimination (IVE), and propose MultiIVE, a new method based on IVE to secure multiple softbiometric attributes simultaneously. Several aspects of this technology are investigated, proposing different approaches to effectively identify and discard multiple soft-biometric attributes contained in face embeddings. In particular, we consider a domain transformation using Principle Component Analysis (PCA), and apply IVE in the PCA domain. A complete analysis of the proposed Multi-IVE algorithm is carried out studying the embeddings generated by stateof-the-art face feature extractors, predicting soft-biometric attributes contained within them with multiple machine learning classifiers, and providing a cross-database evaluation. The results obtained show the possibility to simultaneously secure multiple soft-biometric attributes and support the application of embedding domain transformations before addressing the enhancement of soft-biometric privacy.