Do not underestimate the capabilities of your data migration team. A strong data migration team will likely be willing to help identify issues with the environment builds to aid the overall solution delivery, as well as their own.
Ideally, data quality should be part of a business strategy with defined processes for managing their information sets. However, data quality is often overlooked until a problem arises. In my experience, these problems often bubble up to the surface during data transformation projects. As a result, data quality often becomes an IT issue.
Data transformation is responsible for moving and reporting data from A to B. Generally speaking, most data can be transformed using a fairly simple set of common transformation rules. Very specific and complex business rules are often required to target very specific data inconsistencies. Data inconsistencies that cannot be resolved via business rules are reported to a data quality team to reconciling these errors. Question is, how should these issues be managed? What communication channels are required? How are the issues prioritized? Who is tracking progress and driving progress? Data transformation understands their data requirements. The data quality team knows how to fix the data, but does not know what data should be targeted. The business understands the how the data impacts their business operations.
My experience is that a separate steering committee needs to be adopted so they can provide governance across the three channels (transformation, data quality, and business). The steering committee should be responsible for defining communication tools to track and report data quality issues (may be a web site, spreadsheet, or custom software), working with the business to prioritize issues, establishing timelines for the data quality team, and follow up with data transformation to validate the corrections.