A DSP is typically composed of two processors: a core processor, and an I/O processor. In addition, you usually have some internal RAM. To be useful a DSP core processor works at much higher frequency than the I/O sampling rate. Many DSP algorithms work in the frequency domain. These require frame-by-frame by frame operation, where the core waits for a group of samples, processes them, then writes out a group of samples. Out means to the on-chip memory. In order to be useful, a DSP program must finish in real time before the I/O processor needs to output the next sample. So, a DSP program takes less time to execute than it takes the I/O processor to output a sample. The DSP program receives data in real time too. Depending on frame-by-frame or sample-by-sample mode, the program is interrupt triggered when new data is available. The I/O processor has its own clocks for input and output ports. They can be derived from the core clock or not. It depends on the DSP.
Despite this asynchronous model, the input sampling rate of the I/O processor is very much relevant for some programs that run the core processor. Consider a simple program that adds a reverb. It needs to store some samples, the add them to the output them with some delay. The number of samples you need to wait depends on the sampling rate. So the input sampling rate is a parameter for the DSP program. In order to implement your proposal, you'd need to write a very different program, one that know that every 39/480 input samples must be ignored. The DSP programmers would simply frag the hardware guy that dares ask something like this.
The core clock on modern DSPs is the hundreds of MHz. E.g. this AD
SHARC DSP. The I/O processor on that chip can set the serial ports frequencies using the internal clock, or can accept external clock.
So, what I think of the 1010 board. The "24" clock next to 10K2 is probably 24Mhz, and used through a multiplier to drive the 10K2 core clock in the 100-200Mhz range (rough guess based on the 1000MIPS number I remember the 10K1 claimed). The two smaller clocks to right of it are the I/O clocks. They don't seem to go directly to the 10K2, instead they go to the FPGA which, depending on the sampling rate, selects the right one, applies a divider/multiplier if necessary, and send it to the 10K2 I/O processor clock input pin.
I PMed our local DSP guru (jefemeister) to take a look at my reverse engineering attempt...