Because there are so many different migratory birds, each with their own unique pattern, this is a rather broad question. The answer, therefore, will be based on the general study of migration, rather than individual birds.
Migration is an extremely complex process, and a large amount of research into contributing factors is ongoing all over the world. One essential part of that research is rooted in backyard bird feeding and how it affects migration. Variables are many, and include bird species, age, gender, health, and experience, to name just a few.
To put this large body of work into simple terms, it's generally accepted that birds who plan on migrating will do so whether or not food remains available. The primary migration triggers are change in the amount of daylight, also known as photoperiod; and change in air temperature.
The Cornell University Lab of Ornithology in New York is considered a world leader in the study, understanding and conservation of birds. Your question was posed to them as part of their many interactive bird-watching projects.
From their answer:
Keeping your feeders up has no influence on whether a bird will start its journey south. A number of factors trigger the urge for birds to migrate, and the most significant one is day length. As days grow shorter in late summer, birds get restless and start to head south, taking advantage of abundant natural food, and feeders where available, to fuel their flight.
Hummingbirds are no different from others and will migrate regardless of whether feeders are kept up. However, we encourage people to keep feeders up for several weeks after the last hummingbird leaves the area, just in case a straggler shows up in need of additional energy before completing the long journey south.
Also from their Feederwatch Project:
Some people believe they should stop feeding birds right after Labor Day because the birds’ southward migrations will be interrupted. However, a bird’s migratory urge is primarily triggered by day length (photoperiod), and even an abundance of foods at your feeders will not make a bird resist that urge. In fact, your feeder might provide a needed energy boost along a bird’s migration route.
The National Audubon Society, another world-renowned organization, also addresses the issue, especially for people who do not leave feeders up through the winter.
For people who only feed during the summer, they can stop feeding in the fall whenever they want. Migratory birds leave on their own schedule, regardless of food availability, so continued feeding will not prevent birds from migrating as they should.
Late migrants, young and inexperienced birds and hummingbirds that are not completely healthy may be helped by the presence of your feeder, especially in areas where blooming flowers are scarce in fall and early winter.
Many additional studies indicate that the witness of what seems to be a delay in migration is actually time spent loading up on food and calories in order for the migration to be successful. Some birds actually increase their body weight by 100% before making the trip. The report of one such study, authored by a professor at the University of Rhode Island, can be found here.
For these reasons, I believe that people can leave feeders up all year long without worrying that they're interfering with the natural process of migration.