GSOA incorporates multiple types of omic data at a time. See the paper and documentation to use it in R.

Sorry non-R solution:

With GeneSCF you can keep all the list of genes (multiple files) in a single folder and provide that path in 'geneSCF_batch'. It will perform enrichment analysis for individual list and automatically creats output for every list.

Also choose the tool that regularly update the database/functional information. In that case GenrSCF will give you option of updating the database to current version using 'prepare_database' module before running enrichment analysis.