A frustrating common problem I've been asked about before is how to carve fine lines in traditional hessian backed grey lino blocks without them crumbling.
Here are a few tips and questions to think about to hopefully help solve your problem.
How old is the lino you're using? Ideally lino blocks should be used within a year of being opened. Fresh blocks that are still bendy and have that wonderful linseed oil smell are much less likely to crumble.
Your tools are getting a little blunt. Not only is it easier for your carving tool to slip if they are getting a little blunt, but they are more likely to rip the lino if they are blunt. It's a good idea to hone your tools every 30 minutes or so of carving using a SlipStrop or sending them away to be professionally sharpened.
Accepting that grey lino might not be the best block to carve into for that particular design. When you become more familiar with the types of lino blocks, you get to know which designs will and won't work on them. I've learned, for instance, that if I'm carving a design with lots of intersecting cuts and hash type marks, my grey lino block often can't cope and will start to crumble so I've decided before I even start to use Japanese Vinyl instead of grey lino. Although I prefer how grey lino handles, Japanese Vinyl gives me the flexibility to go a bit wild with my mark making and won't crumble apart.
If the part you're working on has started to fall off but hasn't quite, there is still chance to save it- I've had great results with using a very small amount of superglue (warning that it can get a bit messy) added with a cotton bud or old paintbrush. This will stick that flaking bit of lino back down- leave it for a few minutes to dry fully before continuing to carve though. You do have to have a light touch with this to make sure the 'falling piece' doesn't just stick to your brush though.
Your hands are at an awkward angle. Always try to make the lino block work in the best position for you. Turn the block around so your wrists and fingers can carve these smaller lines freely. You need to be able to move your wrist easily without constraint to achieve neat and accurate lines.
Is your block too cold or too warm? Blocks that are too cold can be hard to carve into and increase the likelihood of your tool slipping. Blocks that are too warm can become very crumbly too so try not to overheat the block. A happy medium I find is using a heated blanket laid on top for a couple of minutes (I always have one to hand anyway when I sit) but a hot water bottle for a few seconds can work well too. Experiment with finding your happy medium of heat using scraps of lino.