Sully is named after the airline pilot Chesley "Sully" Sullenberger, who landed a passenger jet on the Hudson River in 2009, saving all 155 passengers and crew on board.
"As much as our family is going to miss this dog, we're comforted to know he'll bring the same joy to his new home, Walter Reed, that he brought to 41", the younger Bush wrote.
4141 George Bush Locomotive, which was unveiled October 18, 2005, was dedicated to honor the nation's 41st president. The current president gave Bush, a decorated WWII naval aviator who flew several combat missions, a brief salute before the first couple left the Capitol.
He said Bush was the first president he ever voted for and was "the first president to teach me and many of us that in a democracy sometimes you fall short", and how you handle that, "is just as important as how you win".
In Texas, students, staff and visitors have been flocking to Bush's presidential library on the campus of Texas A&M University, with thousands of mourners paying their respects at a weekend candlelight vigil at a nearby pond and others contributing to growing flower memorials at Bush statues at both the library and a park in downtown Houston.
Mammootty Finds a Place In Forbes India Celebrity 100 List!
Aamir Khan, Bollywood Megastar Amitabh Bachchan, Ranveer Singh, Sachin Tendulkar and Ajay Devgn follow next in the top 10 bracket. The Tamil and Telugu film industries have seven actors each on the list while Malayalam star Mammootty represented Kerala.
The Bush family, led by son and former President George W Bush, accompanied the patriarch's body on the flight aboard Air Force One from Texas to Joint Base Andrews outside Washington, and then on a final trip down Pennsylvania Avenue to the steps of the Capitol.
Bush died Friday at his home in Houston at 94.
Trump has ordered the federal government closed Wednesday for a national day of mourning.
Bush will lie in state in the Capitol rotunda for a ceremony and public visitation from Monday evening through to Wednesday.
It features a photograph of Sully laying down in front of Bush and Bill Clinton, the man who defeated him in the 1992 presidential race.
"Our thoughts and prayers are with the entire Bush family during this hard time", said John Miller, president and CEO of America's VetDogs.