As currently written, the question lacks a lot of important information, including the nature and bandwidth of the incoming signal.
That to one side, your maximum sample rate is the vaguely 'a few kS/s'. Let's take that to be 2,500 samples/sec max.
One low-cost solution is a single-chip ADC with 16 muxed inputs running at 1 Msps. That would be used to convert 15 channels in succession, giving 14 us of skew between the first and last conversion.
Since 2,500 sps gives you have 400 us between samples, 14 us of skew is 3.5% error by skew. At your low sample rates, this may be acceptable. Otherwise, you could use two ADCs with 8 muxed inputs at 1 Msps or a single faster ADC.
You could used an microcontroller (MCU) to control them. You can also use an MCU with two ADCs each with 8 muxed inputs. MCU ADCs tend to be lower resolution/quality than the dedicated ICs so you would have to assess the quality available against what you need.