as said above, CX tyres tend to be crap on tarmac, with spaced out knobs affording less grip than a slick tyre, with added squirm, so you just have to go easy into tight road corners
this is tyre tread dependant obviously
off road, they will get sucked into ruts a lot easier than an MTB tyre as they’re a lot thinner so find gaps easier, it’s something you have to compensate for
grip is significantly less than an MTB, and handling is a lots quicker, it’s just something you have to adapt your riding style to, you can’t just blast through stuff, you need to ‘go light’ and ride with a certain amount of finesse, which is what makes it fun
i ride the same South Downs trails on MTB and a CX bike, even the steep ones, it’s the fear on the latter that makes them come alive
hold off on putting the SB8s until it’s completely dry, they’re comically sketchy in the damp :-)