With the availability of modern satellite and data transmittion technology, the possibility of continual tx of FDR information should be possible
The problem is cost, of developing the system, and of transmitting that much data.
I've done some work on airborne satellite data and yes it is definitely very possible using existing services but it's not cheap. Take a big airline, and take how many planes they have airborne at any one time, and multiply this by the cost of (effectively) a DSL connection, and it will be well up in the millions a year.
Some satellite networks are struggling with reliability already. If you loaded up somebody like Immarsat with every jet sending a few hundred kbits/sec the whole lot would just collapse.
Then the data has to be collected from around the aircraft systems. Implementing a certified solution, across the many aircraft types, would be a huge task whose cost would make the satellite data cost pale into insignificance.