I doubt the minimum is rigid and there is a room for lot of considerations.
I'd like to point you to a specific example (IISc). Although the points are opinions of a single person, my conversations with people at IISc have yielded similar opinions.
Prof. Giridhar Madras(Did his PhD from TAMU in <3 years!) has an excellent blog dedicated to IISc/IITs and some excellent articles for prospective faculty.
He points to one page that he authored about recruitments in IISc from which I'd like to quote a few points:
Technically, there is no age bar and no limit to the experience a candidate may have, but with increasing years of experience, candidates must be considered for correspondingly higher appointments. Though IISc has recruited faculty from 27 to 37 years old at the assistant professor level, the median age of recruitment would be around 30.
Begin making your enquiries at least one year before you actually want to move. The time taken to acknowledge an application vary from department to department, depending on the chairperson. If this happens, find someone you may know in the department and request them to check with the chairperson of the department. After an application is acknowledged, try to arrange an informal visit with a seminar. You can indicate that you are planning a trip to India in the month of X and would be happy to give a talk/seminar at that time. In your conversation with the chairperson, state the time frame you want to join the institute. Normally, chairpersons will arrange your talk. Besides the talk, a visit with all faculty will also be arranged. Talk to all faculty and make your research sound interesting to them. Talk about your doctoral and postdoctoral research, your future plans and how you plan to distinguish yourself from your advisors. If possible, explain why you think you will fit into the department. Also, there is nothing wrong in stating that you have applied to more than one place and your preferences depending on the offers.
The suggested timeline is as follows: Apply to several institutes at month X. Follow up and try to get an acknowledgment by month X+1 or X+2. Then, send an email saying that you are visiting India in X+3 month and would like to give a seminar. Try to schedule as many seminars in many institutes in that month. After the seminar, inform them you are willing to return around X+6 months, which will require them to make a quick decision. You can return after X+12 months because the institutes will give you time to join. Please note that each department in IIT/IISc receives at least around 30 to 40 applications per position. Simply applying by email and expecting a detailed response is foolhardy. Unless the applicant shows repeated interest (followed by a visit to give a seminar), it is unlikely that the application will be considered seriously.
The Civil dept at IISc has an informal FAQ for it's recruitments.