The primary distinction is between overturning a conviction on appeal, and exoneration. In the former case, there may be some important evidence that was admitted but legally should not have been: or, defendant's counsel may have been incompetent, and the appeals court overturns the conviction on the grounds that defendant's 6th Amendment rights were violated. Alternatively, one may establish actual innocence, and the defendant may be exonerated (and therefore freed). Typically, official misconduct such as falsified lab reports or withholding exculpatory evidence is assigned to the "exoneration" bucket, since these involve material facts about evidence that were not known at the time (unlike disputes about the law). Either path involves a court hearing.
"Investigation" suggests something substantive regarding the evidence, rather than procedural issues such as wrongly denying a motion to read a particular jury instruction, admitting inadmissible evidence (or the opposite) and so on. In the US, there is an organisation The Innocence Project which focuses on DNA-based exoneration. A similar organization, the Innocence Network, is not DNA-centric. This web site maintains a registry of exonerations, which has numerous case-specific details (this is the place to go if you want the case-specific facts).
The Innocence Project has its own evaluation criteria for deciding whether to take on a case. There is no fixed procedure for deciding whether to investigate, and permission is not required to investigate. Because there are vast numbers of organizations that might be involved in gathering and maintaining evidence in a criminal case, it's impossible to say whether some organization has a special rule regarding access to evidence relevant to a conviction.
At some point, a court order may be necessary, and there are rules about getting one (such an order may be necessary to support a petition to vacate a conviction). An investigation may naturally develop at the hands of the police, for example this case which centered around dirty copy, or this case which involved witness perjury. In this case, a fingerprint analyst came under review, and while the mis-identification did not result in a change of conviction, it inspired the intervention of the Innocence Project, which (legally) aided in DNA testing of evidence, ultimately proving that the defendant was not the source of the DNA at the crime scene.
There may or may not be subsequent investigations when official misconduct is involved – there certainly have been, e.g. the FBI Lab Scandal.
