What happens if any of the rights are abridged in pursuing a conviction?

The general doctrine courts follow is called "fruit of the poison tree."

If a defendant's rights were violated, then any evidence deriving either directly or indirectly from that violation is inadmissible in court. The violation "poisons" the evidence obtained illegally and any other evidence discovered as a result.

It is not, however, a "get out of jail free" card. If the police illegally search your home without a warrant and find heroin there, they cannot use that as evidence. They can, however, use the surveillance video taken of you two weeks ago showing you selling heroin in the park that made them want to search your home in the first place as evidence.