They could be lying but I think some car manufacturers do give more incentives to move cars in different parts of the country.
You are correct; most manufacturers offer different incentives depending on region, or in some cases, metro area. Most VW incentives are national, though they have started introducing regional bonus cash on certain models (to my knowledge, GTI is not currently one of them). The regional bonus cash is based on the selling dealer, so if you're willing to travel, you could save an extra $500 on a Jetta. Nissan, on the other hand, offers varying incentives that depend on where the car is registered, so a customer who lives in the Detroit area may get more cash on an Altima than someone who lives in Cleveland. Toyota incentives are all regional, and their regions are divided along state lines. Audi separate their incentive programs into 4 main regions, though we find that most months, they're largely the same. Ford and GM tend to get geographically specific when dividing up their incentive cash, though GM has recently started offering more VIN-specific incentives, which helps dealers get rid of older inventory.
As to why the DC-area dealers are always advertising lower prices, I'd have to do more research, because I am unaware of any additional regional incentives on GTI, and even if there were any, they'd be available in a wider geographic area. I suspect that the dealers offering the lowest prices are also including some of their marketing allowances and other hidden cash incentives (these are not published on their internal networks and instead communicated directly to their dealers), which they are allowed to do if they choose, much like the national dealer cash program.