Problem with engine cutting out when I have to come to a stop.

What is your question?

Engine idles fine. When driving along, if I have to brake sharply or 50% of the time when I come to a complete stop at a junction, engine will cut out. Always starts straight up and will idle indefinitely, but again, driving along, at the next complete stop, it cuts out. Have changed plugs, leads, distributor cap, in and out of the mechanics a million times and they don't know what the problem is. Now they are suggesting timing chain, or a bad valve. Any ideas anyone?