You would need to track back the US line and confirm as many details as possible on each generation before making the switch to Scottish records - e.g. census returns, birth marriage and death certs, obit, headstones etc. ScotlandsPeople pay-website is good coverage of Scottish records including images of the census returns, civil records, and parish records, but I would wait until you have established a little more information first, otherwise you could use up quite a few credits chasing incorrect lines ..
I had a quick search of US records and found a possible lead...using Ancestry and FindMyPast (Ancestry seem to be having some problems at the moment - some searches were giving inconsistent results) these need to be followed up further to verify the details fit with your Brownlee family...
Starting Richard's father David - he seems to have served in the Military during WW1 as there are various reference to him in these, some of which confirm the connection with wife & children's names, there's also a very likely match for him on the US 1910 census age 16, which fits, at Formosa Street, Pittsburgh Ward 13, Allegheny, Pennsylvania, with three sisters and his widowed mother Annie. David & youngest sister Ruth are listed as born Pennsylvania, Annie snr, Jean and Annie jnr. are listed as born Scotland. I didn't locate any US records which show Annie. snr's maiden surname which would be a vital clue for a search further back... I think establishing that would be the breakthrough to confirming any possible Scottish records...
now for some speculation...
the transcripts on the 1891 Scottish census show a 'Brownlie' household at Crawford Street, Govan, Partick, Lanarkshire, Scotland which might fit - father is David (32) a 'Sewing Machine Mechanic', wife is Annie (28) Annie jnr. (3) and Jane (<1) - Jeane and Jean are interchangeable in Scotland... the ages fit with the 1910 US census record..
and finally the UK kept some records of passengers leaving from the 1890s onward and a search of these show a Brownlie family leaving Glasgow, Scotland for New York, USA in 1893 on the Devonia (Anchor Line) - with David a farmer[?] wife Annie (no ages given), children Anne (4), Nellie (2)[?] and Jannie (infant)...