My infield dirt has, over time, become much lower than the infield grass. I'm referring to the edge where the dirt meets the grass on the infield side (closest to home-not the back edge where it meets the outfield grass). At some points it drops as much as 3". The cause is not a "lip" in the grass edge the infield is relatively flat. I suspect the dirt has eroded to the back edge with water/rain drainage towards the outfield, but I am not sure.

An example to clarify what the problem is: Imagine you're playing third and the batter hits a slow roller. You charge the ball, but stump your toe and fall on your face. Looking back you realize you have tripped over the grass edge of the infield.
This could easily happen.

Anyone dealt with a problem like this? Any relatively easy remedies?

Low cost rent a dethatcher put it on the lowest setting and walk the infield. We do this every year right before we over seed and it work great.
The other thing you could is rent a sod cutter, cut into the infield pull it back, take out some dirt and relay the sod. Most expensive bring in new dirt. I have done all three and they all work.

Before you do anything look to see if the dirt has just moved to the back of the IF towards the OF grass. If it has here is a pretty simple solution.

Take a disc (or disk - not sure how to spell it) and dig deep into the dirt near the outfield. Then push the dirt forward to the lip and spread / smooth it out. You can push it using field rakes (good workout for the guys).

Once you get the dirt how you like it then take a heavy roll over it to compact it.

Ryno;
I've got, and had, the same problem. I think you just need to have a load or two of dirt delivered and dropped on the infield and then use a blade or someone good on a bobcat to level it out.
As a matter of fact I was just surfing for places to buy infield material when I saw your post.

This is a common problem on baseball fields where the infield clay/dirt migrates in under the edge of the grass. You can minimize it by being careful when you drag to not drag toward the grass and be careful around the edge. Get a large push broom and have your kids brush the grass toward the dirt about a foot in the grass after every game and pratice. In addition if you hand water the grass take a stream of water outward from the grass toward the dirt. All of these things will help minimize it. When it gets too bad then get a sod cutter and cut it out, re-level the area and replant or resod.