I ran into a problem where I noticed that when in the big chainring (53t ultegra), and shifting to the lower end of the cassette (19t or 21t for example), my chain has a tendency to drop down to the smaller chainring . I know that cross chaining isn't advisable however this is still a few cogs from the lowest gear. Is this possibly a chainline issue or are there any other things I can try?
One other thing to check is how the chain feeds onto the chainring. The chain should be able to stay on the chainring coming from any cassette position except maybe the most inboard one or two (max.). If the chainline is per spec, trim the FD so it doesn't touch the chain at all, and watch to see how the chain is feeding onto the chainring. Odds are it's catching on the first tooth after the shift gate, but it could be catching on any tooth. that;s because when the chain feeds from an angle the upcoming teeth touch the inner plates rather than finding the gap between them.
There are fixes. One is to find a chain with more bellmouth formed into the inner plates, forming a wider funnel to meet the teeth and guide them in. Another is to move the points of the teeth, or at least the one after the shift gate more inboard. You do this by filing the outer tip at the same taper angle as now, so the point is a bit more inboard. For all the teeth you can lay the file against them and spin the crank, for just one, use a small file and do a light trim.
Before you try a radical solution, make sure you've exhausted all others. If it's just marginal for example (your's seems worse) sometimes better chain lube helps. One thing LBS mechanics do is trim the FD so the cage pushes the chain over and guides it on. I consider this unacceptable since chain tension will strain the FD when climbing, and eventually the chain will saw it's way through the cage.