It should be kept in mind that peer review is not designed to ferret out fraud, nor is it about replication, nor is it about deciding something is "true." It is about making sure that some basic disciplinary norms have been met, and making sure things pass the "smell test" of an interested observer who knows a lot about the subject matter, but doesn't proclaim to be an expert in that specific field. This is as true in the sciences as it is in any field. People often put a lot on peer review's plate that it can't handle! Many of the functions they ascribe to it are actually done by other activities in the discipline, if they are done at all. (E.g., when historians disagree about sources/conclusions, they don't do this through anonymous peer reviewing so much as separate, non-anonymous publication. Imagine how burdensome peer review would be in science if it required replication!!! Consider how impossible it would be every study needed to be done twice before its results were published!)

As for whether sources are consulted by peer reviewers, it can happen but it's going to depend very much on the reviewer. Nothing I've ever had peer reviewed suggested people looked up sources. I have done it on a few occasions but only when I'm reviewing things that pertain to what I directly work on, and when the sources are already at hand.

Historians often do track down citations of others, check if they say what they are claimed to have said, look at the full context, even see if they exist, etc. But this does not happen at the peer review level.

It's pretty tough. There are cases, like with Michael A. Bellesiles, where peer reviewers didn't look at the sources used within the work. In Bellesiles's case, he won the Bancroft Prize before anyone found out his sourcing was dubious. This case sparked a lot of debate about peer reviewing, and questioned how peer reviewers would go about checking sources, especially obscure and hard-to-obtain sources, like those used in Bellesiles's book "Arming America: The Origins of a National Gun Culture." There still isn't really a solution. Prestigious prizes, like the Bancroft, have started to look into the sources of their nominations, but it's just too tough for a lot of regular peer reviewers to do alone, and takes up far too many man hours.

It's best to have someone within the sub-field of the work peer review it, but even they won't have access to all of the records. For instance, Bellesiles claimed to have studied like 10,000 court records from all around the country, and these records were created in the late 18th century. Until the majority of these records are digitized, there's just far too much work to verify any single claim made in a historical work.