Hi Evie
I guess not every single passenger list is available on line, though the Ancestry database for example, is ever increasing. And with any country's Regional archives etc, I would imagine there would be some Manifests at least which are unreadable, or missing at least some pages of....
Per the UK Census: Thomas was a Coal Miner, and the children's ages in 1851 were : Robert 9 (YOB abt 1842) , Mary 7 (YOB abt 1844) , Phoebe 5 (YOB abt 1846) . Plus there was dau Grace 11 (YOB abt 1840) - did she not go the US with the family?
three things do come to mind :
one, Thomas BOOTH very possibly went to the New Land ahead of his wife and children by at least some months or weeks, probably (but by no means certainley) no more than a year at the most - this was very very common and probably so that Father could arrange or establish work and housing before the family arrived. Depending on the age of his son Robert, if he was older, say over 15 or so, it is possible Robert could have accompanied his father in this manner.
two: They may not be listed by their full forenames on the manifest. Thomas BOOTH could be listed as Tho, or Thos BOOTH. Wife Ann could be just Mrs Booth and the children simply as child or infant. They could all or some just be listed simply by their fore-initials and a surname.
three: in order to further narrow down the timeframe, can you find any of the grown children as (married?) adults in the later US Census, the ones which hsave provsion to state the year of immigration for people born outside of the US?
Having said all that, however, there is this passenger list - sans Thomas - which looks distinctly possible to me : Right set of names in the right order, right time frame and entry into the southern states:
1857: "Ironsides" , departed Liverpool, arrived New Orleans, Louisiana on 17 Nov 1857
Ann BOOTH (f) 38 - YOB 1819
Robt BOOTH (m) 16, YOB 1841
Mary BOOTH (f) 11, YOB - 1846
Foley? BOOTH (m) 7 - YOB - 1850
Ann BOOTH (f) 4 - 1853
The ages of children Mary and Ann are a bit out , but not wildly so - and is the male "Foley" really the list maker trying to work out female "Pho(e)be, also with the age a bit out? Perhaps he was copying from an initial list onto the arrival manifest.
Ages were often out on Ship manifests for various reasons: rounding off, illiteracy, mistakes, or flouting of immigration rules which were in place at the time etc. And, it may be due to how accurate the 1851 census ages were , or when they had birthdays, and perhaps if they had a birthday due or just passed, or had sometime between embarking and disembarking.....for example
If this is the wife and children - I bet it is - we could be looking for Thomas arriving sometime before them from around late 1856 to Nov 1857 or so - and not necessarily, but probably departing Liverpool and possibly entering via New Orleans or PA? He could tho, have came into NY or MA and made his way down therefore causing the wife coming into a port closer to their new home.
Cheers
AMBLY