After a while and By and by

By and by and after a while are semantically related in happening topic. In some cases you can use "By and by" instead the phrase "After a while" as an adjective or an adverb, when it comes to topics like later, eventually.

By and by adverb - At or within a short time.

After a while and by and by are semantically related in happening topic. You can use "After a while" instead the phrase "By and by" as an adjective or an adverb, if it concerns topics such as later, eventually.