Gun control is one of those issues where candidates don't like to be pinned down. If they make a specific proposal, the NRA will jump all over them, so they talk in generalities, as Clinton did this time. It's better to avoid getting specific because voters can ignore the gun control issue and vote for or against you based on other things. My specific proposals would be:
1. Expand background checks to make them universal, so they cover gun shows, private sales through the want-ads, everything. This does not violate the 2nd Amendment, and the NRA has a difficult time opposing it.
2. Figure out a way the federal government can dilute or negate the "stand your ground" laws passed by so many states. These laws are dangerous mostly because they're so misunderstood. Many people think a SYG law means, "I can shoot you if I see you on my property."
3. Enact consumer protection legislation to protect people from cheap firearms, usually called Saturday Night Specials. It is perfectly reasonable to set minimum standards for reliable function, safety, etc. This would get the cheap knockoff firearms off the market. If you have to ask what good that would do, you probably don't know about the millions of Chinese copies of the AK-47 rifle and Uzi submachinegun.
4. Safe storage of firearms to discourage children from playing with guns.