It is shown that a mechanism involving only random motion of the sun with respect to the surrounding star field can account for the ~1 per 25 Myr characteristic frequency of large cometary impacts on Earth. In the proposed mechanism, the sun travels through the Oort Cloud of an encounter star, most typically a Type M dwarf, while the dwarf flies through the Oort cloud of our Sun. As a result, Oort Cloud objects from our Solar System are precipitated in large numbers to impact planets in the dwarf star system, while the dwarf's Oort Cloud objects are destabilized to impact planets in our Solar System. It is shown that it is this exchange of Oort cloud object between stellar systems, rather than the precipitation of Oort Cloud objects within a stellar system, that can account for the apparent periodicity of mass extinctions. Because the sun is more massive than ~90% of stars, its Oort cloud extends further, resulting in it delivering about a factor of three more bombardments on other solar systems than our Solar System receives. About 60% of the bombardments on our Solar System are found to be delivered by Type M dwarfs, about 20% by type K dwarfs, with the remaining 20% being delivered by stars of type G or larger. Foreign star Oort cloud objects can be captured by our Sun at typical ranges of 10 AU, resulting in a cometary approach to perihelion within a few years. It is found that assuming an effective Oort Cloud radius of 40 000 AU for a star of solar mass, increasing in size with the square root of the mass, accounts for the observed characteristic frequency of mass extinction events on Earth, given the local stellar number density of 0.003 stars per cubic light year. The frequency of mass extinction events in other solar systems would increase or decrease in linear proportion to the local stellar number density. It is shown that this exchange of materials between solar systems during close stellar encounters could be an important mechanism for spreading life throughout the galaxy. Implications for the evolution of life on Earth and in other solar systems are discussed.