The prior owner of a dog cannot normally be held responsible for harm caused after ownership is transferred, provided that he retained no further interest in the dog and did not misrepresent its temperament or warrant that it would not create the harm in the future. For an excellent discussion of various possible causes of action based on harm occuring after transfer of ownership of a dog, see Blaha v. Stuard (2002) 640 NW 2d 85 (South Dakota Supreme Court).

Rescues and animal shelters often misrepresent the temperament of dogs. Additionally, these organizations and agencies even go so far as to warrant dogs as being safe despite knowing that they are prone to attack people. An expose by Colleen Lynn of DogsBite.org provides vivid examples of what amounts to criminal fraud in the practices of a government-run animal shelter in California. See What's Behind the Click and Bait Web Advertisements of Aggressive Shelter Dogs Available for Adoption Today? An example from that article:

Post regarding Rainbow, a vicious pitbull which was adopted-out by a government shelter.

The motives for releasing vicious dogs to unsuspecting families with children range from overzealous commitment to preventing the animals from being euthanized, to slavish devotion to maintaining the appearance of being a "no-kill" shelter. The methods used for tricking people into adopting unadoptable dogs include misuse of temperament tests (see Alexandra Semyonova, Behavior Testing Shelter Dogs -- The Reality of Where We Are Now) and "dog laundering" which is the term coined by Attorney Kenneth Phillips to describe the practice of moving a dog from one group to another for the purpose of sanitizing its record of vicious behavior toward people and animals. (See Phillips, Don't Support Dog Laundering.) In some cases, vicious dogs are stolen from shelters so that they can be released illegally into the community or even adopted out to a family. (See Associated Press, More dogs stolen from Columbia animal shelter.) In one notorious case, a dogo Argentino that killed a man in New York was released for rehabilitation to a rescue group in Ohio, and then was stolen. (See Jim Sielicki, BLADE, Dog freed in murder stolen from local kennel.) 

The following animal shelters have recently been implicated in "dog laundering" schemes to re-home vicious dogs with unsuspecting families:

The consequences of re-homing vicious dogs are often drastic. On April 29, 2016, a pit bull-mix rehomed by the San Diego Humane Society killed a baby, 3-day old Sebastian Caban. (See Colleen Lynn, 2016 Dog Bite Fatality: Pit Bull Rehomed by Humane Society Kills Newborn Baby.) Temporary fosters working with adoption groups and rescue groups have been killed and injured. (See Rebecca Carey, Georgia Student, Killed By Dogs She Rescued.) Dog bites increase in frequency throughout the geographic area. (See Dog Bites Increase 35% in Austin After the Adoption of 'No-Kill' Policy.) The acting director of the Fairfax County Animal Shelter resigned during an investigation of its adoption practices. (See WUSA9, Multiple dog attacks inside animal shelter.) Adoption groups, rescues, and governmental animal shelters can be prosecuted criminally and sued for damages. (See the Jennifer Lowe Case and the Krystal Cooney case, both of which were handled by Attorney Kenneth Phillips, at Animal Control Liability for Dog Bites.)