The DMM has a bandwidth of maybe 100 kHz, the scope 50 MHz. The bandwidth is about 500 times higher, so as first approximation (eg. white noise), you might expect the noise to be ~25 times higher (sqrt(500)). The exact figure of course depends on the power spectral density of the noise (it's unlikely to be flat up to 50 MHz). Noise is usually quoted for a certain bandwidth, eg. 1 MHz. If it's a temperature sensor, for example, you're unlikely to care about 10 MHz signals. You could add a low pass filter between scope and DUT, or even in the DUT itself.
The scope will also have a fair amount of inherent noise. To estimate it, you could short the probe at the tip (don't change the loop area between probe tip and ground lead) and measure the noise. The apparent noise (as shown on the scope) is the root of the sum of squares of the scope's noise and real DUT noise, both as RMS voltages. This assumes that these two noise sources are uncorrelated.