The point here is that you want to remove the food from your mouth without making a big display of half-chewed food to other people. So if it's gristle, fat, or otherwise gross-looking, discreetly get it into a napkin. Wipe your mouth and spit while doing so. Then close up the napkin (don't leave it open for all to see the contents) and sit it next to your plate, or bin it if possible. Regurgitating it onto a fork is much more visible and not always the best way to go. I've seen far too many people 'miss' the fork when trying this. Even if they don't, the resulting mass on the fork is unpleasant. Be courteous of other diners and just use a napkin.
If it's a bone that you didn't realise was in the mouthful of food, it can sit on the plate if it's clean (not covered with bits of fat or meat). A clean fish bone, for example, isn't gross or off-putting. And at some of the meals I've been to, if you had to put fish bones in napkins each time you found one, you'd run out of napkins. Those little buggers can hide very well in the meat no matter how hard you try to avoid them or remove them in advance. Use a hand to take it out quickly and without fuss.
Don't try to swallow. If you can't chew it into a swallow-sized piece, then it's simply asking for trouble.