when i say dsp it potentially includes crossover and room correction.
my personal dsp tool is my Trinnov, which certainly does both effectively. up to 64 channels. personally i do not use the crossover capabilities. only the room correction, Optimizer and speaker location re-mapping. the Trinnov would do the crossover internally, convert it to analog, and output direct to the driver.
as far as a digital crossover for 2 channel, it would seem to me to be ideally in a one design dac/crossover/speaker such as the Goldmund. so you avoid the additional conversion step. this where the Trinnov excels with highest end Home Theater speakers with bi-amp and tri-amp choices. it can take a digital source, do the dsp work and then convert that to analog direct to the speaker driver. so keeping it digital until it's given to the driver. i suppose most of those might still have a passive crossover circuit of some sort involved. but really have not looked close enough to know.