Elizabeth (mother), David, James (known as Thomas), Jane, Mary Anne and Frederick - (Thomas the father had died by the time they emigrated in several stages).
(excerpt of RootsChat message)

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There is at least one record of a KENYON family living near Nelson in the 1860s:

Nelson Evening Mail
August --- 1868
births:
KENYON, 25 Aug, at Dovedale, Mrs H Kenyon, a daughter

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Emigration date included in J T Norcross military record.

Piece: RG9/3142 Place: Alston -Lancashire Enumeration District: 4
Civil Parish: Dilworth Ecclesiastical Parish: Ribchester Church
Folio: 54 Page: 12 Schedule: 58
Address: 2 Willows

Surname First name(s) Rel Status Sex Age Occupation Where Born Remarks
KENYON Alfred Head M M 36 Stone Mason Lancashire - Dilworth
KENYON Elizabeth Wife M F 35 Lancashire - Dilworth
KENYON Alice E Dau U F 10 Scholar Lancashire - Dilworth
KENYON Margreat Dau U F 11 Scholar Lancashire - Dilworth
KENYON Jane Dau U F 7 Scholar Lancashire - Dilworth
KENYON John Son U M 2 Lancashire - Dilworth
KENYON Mathew Son - M 3d Lancashire - Dilworth
FLETCHER Alice Servnt W F 59 House Servant Lancashire - Grimsargh
FLETCHER William Lodger U M 22 Stone Mason Lancashire - Dilworth

Lancashire OnLine Parish Clerks
Baptisms: 13 Apr 1851 St Wilfrid, Ribchester, Lancashire, England
Alice Elizabeth Kenyon - D of Alfred Kenyon & Elizabeth
Born: 31 Mar 1851
Abode: Longridge Dilworth
Occupation: Mason
Baptised by: Wm Maude Haslewood Curate
Register: Baptisms 1813 - 1851, Page 296, Entry 2368
Source: LDS Film 94020

from LancashireBMD
Surname Forename(s) Mother's Maiden Name Year Sub-District Registers At Region Reference
KENYON Alice Elizabeth BIBBY 1851 Alston Preston Lancashire ALST/4/92

New Zealand BMD

Registration Number Bride's Given
Name(s) Bride's Family
Name Groom's Given
Name(s) Groom's Family
Name
1911/5541 Alice Elizabeth Norcross Henry James Bradley

NZ BMD:

1935/19177 Bradley Alice Elizabeth 84Y

LancasterBMD has record that seems the most likely:
Lancashire Marriage indexes for the years: 1875
Surname Forename(s) Surname Forename(s) Church / Register Office Registers At Reference
NORCROSS Thomas KENYON Alice Elizabeth Longridge, St Lawrence Preston 243/1/160
Also, the groom THOMAS matches better with census records. (FamilySearch missed it as it has lastname under Norcufs)

Groom's Name: Thomas Norcufs
Groom's Birth Date: 1851
Groom's Birthplace:
Groom's Age: 24
Bride's Name: Alice Elizabeth Kenyon
Bride's Birth Date: 1852
Bride's Birthplace:
Bride's Age: 23
Marriage Date: 22 Jun 1875
Marriage Place: Longridge, Lancashire, England
Groom's Father's Name: John Norcufs
Groom's Mother's Name:
Bride's Father's Name: Alfred Kenyon
Indexing Project (Batch) Number: I04110-0
System Origin: England-EASy
Source Film Number: 1657578
Reference Number: item 160 p80
Collection: England Marriages, 1538–1973

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1881 Census

Name: Thomas Norcross
Age: 29
Gender: Male
Birth Year: 1852
Birthplace: Ribchester, Lancashire, England
Relationship to Head of Household: Head
Marital Condition: Married
Profession/Occupation: Timber Merchant
Address: Lumb Mill
Census Place: Dilworth, Lancashire, England
Disability:
Record Type: Household
Family History Library Film: 1342016
The National Archives Reference: RG11
Piece/Folio: 4245 / 91
Page Number: 15
Household Gender Age
Thomas Norcross M 29
Spouse Elizabeth Alice Norcross F 29
Child David Kenyon M 7
Child Margaret Norcross F 5
Child Elizabeth Norcross F 4
Child John Norcross M 3
Child Richard Norcross M 2
Child James Thomas Norcross M 1
Child Frederic Norcross M 0

============

another version of the same data:

Household Record 1881 British Census

Name Relation Marital Status Gender Age Birthplace Occupation Disability
Thomas NORCROSS Head M Male 29 Ribchester, Lancashire, England Timber Merchant
Elizabeth Alice NORCROSS Wife M Female 29 Dilworth, Lancashire, England Housewife
David KENYON Son U Male 7 Dilworth, Lancashire, England Scholar
Margaret NORCROSS Daur U Female 5 Dilworth, Lancashire, England Scholar
Elizabeth NORCROSS Daur U Female 4 Ribchester, Lancashire, England
John NORCROSS Son Male 3 Gisburn, York, England
Richard NORCROSS Son Male 2 Dilworth, Lancashire, England
James Thomas NORCROSS Son Male 1 Dilworth, Lancashire, England
Frederic NORCROSS Son Male 1 m Dilworth, Lancashire, England

Source Information:
Dwelling Lumb Mill
Census Place Dilworth, Lancashire, England
Family History Library Film 1342016
Public Records Office Reference RG11
Piece / Folio 4245 / 91
Page Number 15