They can all describe the same action but have slightly different meanings.
When you are traveling somewhere, you can swing by another place by going a little out of your way to stop there for a while.
I can swing by after work.He swung by here on his way home.
When you are traveling, you can stop by a place by visiting it.
I can stop by after work.He stopped by here on his way home.
When you are traveling somewhere, you can drop by another place by visiting it briefly.
I can drop by after work.He dropped by here on his way home.
So as you can grok from the examples, they’re very interchangeable. The difference that exists is in emphasis. “Swinging by” focuses the imagery on the deviation from one’s main route and thus on how the person will eventually need to get going and reach their final destination. “Dropping” and “stopping by” are mostly synonymous, focusing their imagery on the visit itself, but the latter implies a longer visit (or greater willingness to stay longer if needed) than the former.
But, of course, they’re all somewhat informal so people aren’t usually using them with great care; the nuance among them will vary from person to person and from context to context.
Edit: At least one other internet conversation about this includes a Brit’s opinion that “swing by“ is an Americanism—it doubtless derives from swerving one’s car rather than using a vine—and several people’s feeling that they use “drop by” more in reference to calling on someone than, e.g., picking up something at the supermarket. That hasn’t been my experience, but that’s how they use it. I would take “come by” that way, though, since you’re obviously mentioning it to someone who will be waiting there for your visit. Some other discussions here, here, and here, all basically finding that they're mostly interchangeable. One person felt “swing by” intended a briefer visit; another inexplicably felt “stop by” was shorter; a few felt “stop by” was slightly more formal and more appropriate in business contexts.
