According the the nfl.com website once a field goal is missed the other team takes control of the football no matter what down it is.
Here's the quote from nfl.com:
All field goals attempted (kicker) and missed from beyond the 20-yard line will result in the defensive team taking possession of the ball at the spot of the kick. On any field goal attempted and missed where the spot of the kick is on or inside the 20-yard line, ball will revert to defensive team at the 20-yard line.
Doesn't mention anything about downs. No. The reason you might sometimes see a team try for the field goal on 3rd down is in case of a bad snap or something they can spike the ball, throw to a receiver or just run with it and still have fourth down to try the field goal. If the ball fails to cross the line of scrimmage and the offense recovers, they get to kick it again. This usually only happens on a blocked kick. This applies on any down other than 4th. If the offense recovers on 3rd down, the next play would be 4th down. If the offense recovered on 4th down and didn't gain the necessary yards to make a 1st down, it would be a turnover on downs.
The above answer is correct. As long as the ball does not cross the line of scrimmage on a blocked kick and the offensive team recovers the ball, the team may attempt another field goal as long as the blocked attempt did not occur on fourth down.
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