The UR series, as any interface in comparable price ranges, have at best 20 bits of usable data (actually rarely more than 18 bits), the rest is just electronic noise. The best ($$$) interfaces barely make it to 21 bits, the rest again is electronic noise.
It would be a (very!) long explanation involving laws of solid-state physics, but it’s very hard to do any better at normal ambiant temperatures, and running your device at cryogenic temperatures would be “impractical”… 
So the 32-bit figure advertised by Steinberg is 100% marketing fluff (to stay polite). Running at 32-bit mode instead of 24-bit mode just gives you 8 more bits of useless noise… pure waste.
You do have some gain (about 2-3 useable bits) when using 24-bit mode instead of 16-bit mode, but no value whatsoever to move up to the 32-bit format. (for recording that is)
A few rare companies (ex. SD Mixpre) use a 32-bit float mode combined with 2 (or more) AD converters in parallel. The clever trick (one way to do it) is two have 1 ADC tuned for lower-level signals plus 1 ADC tuned for higher-level signals. An FPGA (typ.) then selects the data from the first one as long as it does not clip, then switches automatically to the other one when the signal gets hotter. The numerical conversion is done on the fly (typ. FPGA) transparent to the user, except a slight latency penalty… well… there is also a significant penalty on your wallet $$. (other DSP-based variants also possible, but basic idea is similar)
In any case, you still don’t have a true 32-bit dynamic resolution, but you get better headroom and immunity to clipping.
You don’t need a pricey interface. The best solution is still simply to lower your faders to stay at a clear distance from clipping.
By definition, you will always clip when reaching 0 dBFS, whatever the resolution used.
Sorry for the long explanation…
Short version:
If you measure your Christmas tree in millimeters instead of centimeters, you will have a better resolution/precision, but it will still be too large (clip) for your car, whatever the units…