Moving between different industries in private industry?

How important is choosing the right industry at the staff accountant position? I am currently searching for an opportunity in private accounting and trying to get a sense of whether a certain company/industry will pigeon-hole me to a specific industry. From my understanding, general accounting (ie: posting entries) seems to be highly transferable. However, I have heard that Fund Accounting is quite specialized. What is the consensus on private accounting in healthcare, financial services, technology, entertainment, etc.? Please let me know your thoughts!