Word 2013 will open Word 2003 files, so there is no great need to convert them (if they don't open, check your file block settings). Converting them can produce formatting anomalies. However if you feel you must convert them, http://www.gmayor.com/document_batch_processes.htm will do the job (subject to the same potential file block issue).

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Graham Mayor - MS MVP (Word) (2002-2019)
Visit my web site for more programming tips and ready made processes www.gmayor.com