Hi Debbie!
Here is emigration records which I found on the Emibas CD, this CD covers about 75% of the people who were leaving their parish to emigrate to a forign country.
The names in this family match the ones you wrote about above except I have not been able to locate anyone with the name Mathilda and I have not been able to find where this family returned to Sweden and then emigrated a second time in 1882.
I have also located the family in the Clergical Survey records (HFL), The family can be found at AI:56, 1876-1882, Norra Förstaden 36, Landskrona, Malmöhus län, page 433. Genline ID: 1292.30.50900
Here are the records I found on the Emibas CD:
Post 475925
Andersson, Nils
Snickare (married man)
b. 7/10/1839 in Billeberga, Malmöhus län (Skåne)
Emigrated 5/25/1882
from Norra Förstaden 36, Landskrona, Malmöhus län (Skåne)
to Amerika
Source: Household Examination Roll, p. 433
Emibas migration file ID: Landskrona M 1882 073
Post 475926
Persdotter, Kjersti
Hustru (married woman)
b. 7/18/1833 in Nöbbelöv, Malmöhus län (Skåne)
Emigrated 5/25/1882
from Norra Förstaden 36, Landskrona, Malmöhus län (Skåne)
to Amerika
Source: Household Examination Roll, p. 433
Emibas migration file ID: Landskrona M 1882 074
Post 475929
Andersson, Ida Maria
Dotter (unmarried woman)
b. 3/24/1863 in Landskrona, Malmöhus län (Skåne)
Emigrated 5/25/1882
from Norra Förstaden 36, Landskrona, Malmöhus län (Skåne)
to Amerika
Source: Household Examination Roll, p. 433
Emibas migration file ID: Landskrona M 1882 077
Post 475930
Andersson, Anton Wilhelm
Son (unmarried man)
b. 3/30/1865 in Landskrona, Malmöhus län (Skåne)
Emigrated 5/25/1882
from Norra Förstaden 36, Landskrona, Malmöhus län (Skåne)
to Amerika
Source: Household Examination Roll, p. 433
Emibas migration file ID: Landskrona M 1882 078
Post 475928
Nilsdotter Andersson, Emma
Dotter (unmarried woman)
b. 2/29/1867 in Landskrona, Malmöhus län (Skåne)
Emigrated 5/25/1882
from Norra Förstaden 36, Landskrona, Malmöhus län (Skåne)
to Amerika
Source: Household Examination Roll, p. 433
Emibas migration file ID: Landskrona M 1882 076
Post 475927
Nilsson Andersson, Johan August
Son (unmarried man)
b. 7/1/1871 in Landskrona, Malmöhus län (Skåne)
Emigrated 5/25/1882
from Norra Förstaden 36, Landskrona, Malmöhus län (Skåne)
to Amerika
Source: Household Examination Roll, p. 433
Emibas migration file ID: Landskrona M 1882 075
Hope this helps!
Have a nice day!
Don