If you're looking for all of the children of William Crawford and Martha Moody, I might be able to help. I'm descended from their son William (1865-1934).
In total, William Crawford and Martha Moody had 10 children.
To my knowledge, none of their children moved to England.
One son did move to New Zealand.
Andrew Crawford (1851-1933) m. Mary Wham (from Illinois)
Andrew emigrated to the USA on August 21, 1871. They had 5 children. 3 sons and 2 daughters.
James Crawford (1853-1928)
James moved to New Zealand. I have been in contact with his descendants and am adding their information to my family tree. There are 3 generations after James' arrival in New Zealand.
John Crawford (1855-1914) m. Margaret Hutchinson (1878-1941)
This family emigrated to Canada. They had 7 children.
Margaret Jane Crawford (1858-1938) m. Frederick Nordeck (c1870 Iowa City, Iowa - ?)
Margaret emigrated to the USA on May 22, 1885.
Mary Crawford (1861-1865) She drowned as a child.
Henry Crawford (1862-1933) m. Sarah Pollock (1864-1939)
Henry stayed in Ireland and farmed the Crawford family farm in Glebe, Co. Londonderry, Northern Ireland.
I have another 2 generations of Henry's family in Northern Ireland if someone wants more information.
William Crawford (1865-1934) m. Ada Dianne Homer (1867-1938)
This is the branch of the Crawford family tree that I am descended from. William and Ada are my 2x great-grandparents. William emigrated to Canada at the age of 18 (in 1883) to work as a farm labourer. William and Ada settled in Ontario near Newmarket, Ontario and had 6 children.
Joseph Crawford (1867-1868).
Joseph Crawford (1870-1937)
He never married. He was a tailor by trade. He is buried in Omagh, Co. Tyrone, Ireland.
Martin Crawford (1872-1951) m. Margaret Hutchinson (1878-1941).
Martin emigrated to the USA with his sister Margaret and his brother Andrew. He emigrated on October 15, 1897. He settled in Iowa City, Iowa.
He later moved to Canada in 1925 and married his brother John's widow Margaret Hutchinson.
As for the Crawfords who emigrated to County Down, UK the only thing I can think of is that they may be descended from another branch in the Crawford family tree. I can trace my family tree back one more generation from William Crawford and Martha Moody to William's father - James Travers Crawford Senior and his wife Margaret Hunter. None of their children emigrated anywhere other than Iowa, USA...if they left Ireland.
Hopefully, that information helps!