What did a semen-stained blue dress from the Oval Office have to do with a land deal in Arkansas? After a couple of years of digging, that's all a partisan prosecutor could find against Bill Clinton. 
There's nothing unconstitutional about a prosecutor working an investigation finding evidence of other, related or even unrelated crimes. And of course, it's constitutional to pursue those other crimes. 
Watergate started as the investigation of a burglary and led to related crimes that brought Nixon down. 
Say a judge signs a search warrant based on evidence of an illegal poker game in Joe's basement. The cops come in and find they're not gambling, but there is an ounce of cocaine on the card table. Do they walk out and say: "Darn, too bad the warrant's for illegal gambling. Now we can't make a case for the cocaine."
Same as a prosecutor investigating MAFIA underlings to build a RICO case against the mob boss. She may use a drug case as a means of gaining cooperation.