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I'll mention the following three things for you to consider.

Firstly, the "scotlandspeople" baptism index shows a Catherine Murray baptised (RC) on 12 April 1838, parents recorded as John Murray and Matilda McHaffie, the parish was Glasgow, St. Andrew's. I couldn't readily see a marriage for them.

Secondly, there was a marriage in St. Peter's RC Church in Belfast on 30 March 1900 between a widowed bootmaker called Thomas Murray, son of a shoemaker named John Murray, to an Ellen Savage:
https://www.irishgenealogy.ie/files/civil/marriage_returns/marriages_1900/10342/5765230.pdf

On it's own, of no great import... except that the RootsIreland subscription website has a transcript of the church record of that same marriage which also identifies Thomas' mother - namely one "Matilda Mahaffey". When I go looking for Thomas a year later in the 1901 Ireland Census, the following looks like a possibility, with Thomas' birth indicated c. 1846 in Scotland:
https://www.census.nationalarchives.ie/pages/1901/Antrim/Falls/Cyprus_Street/951042/
 
The Thomas in that census return died on 7 December 1908 with a reported age (by his wife) indicating birth c. 1851:
https://www.irishgenealogy.ie/files/civil/deaths_returns/deaths_1908/05481/4534962.pdf

Thirdly, the following item in the Newry RC registers is difficult to read, but if you fiddle with the light and contrast, you should be able to make out the baptism of a Jane Murray on 28 February 1848 for parents John Murray and Matilda Mehaffy, sponsors Thomas [something] and Mary Gallagher:
https://registers.nli.ie/registers/vtls000633289#page/31/mode/1up (left page)

Forgive me if I am misunderstanding, I'm so excited to have had replies that I may be rushing through reading.

 I had suspected, based on the existence of records indicating another McHaughey/Treanor/Murray convergence in the same area as my ancestors in Glasgow that perhaps they had had pre-existing connections there before they moved there themselves.
I thought maybe they were unconnected and I had stumbled upon them simply by virtue of looking specifically for those names, but the physical proximity did give me pause. Would these records indicate to you that perhaps John and Matilda McHaughy/Mahaffy/whichever spelling were themselves from Scotland? I had thought maybe that during the famine years they had maybe been travelling between Ireland and Glasgow for work, given that there can't have been much to go around for a shoemaker, but I am very very much an amateur and have nothing but my own speculation to back this up.

Thank you so so very much for replying!

I am looking for any information/tips for where to search next for ancestors by the name of Murray who lived in Newry at least as early as 1840, and in my direct line had emigrated to Scotland by 1898. The last date of residence in Newry I have for certain is 1894, as per the record of death for their second eldest daughter, Catherine. I have managed to find quite a lot more than I expected, as I had only a name and birth place for my great grandather's father, but I have hit a wall around 1840 and I am wondering whether there is anywhere else to look, or if I should just accept that the records are either lost to time or never existed. Here is what I have so far:

Griffith's valuation places a John Murray in High Street, Newry, in tenement housing (I believe in 1846?). There is a James Murray born in August of 1844 to John Murray and Matilda Haffy in the diocese of Dromore, followed by a sister Matilda on July 19th, 1846, and potentially a Robert in 1850, and Patrick in 1853, although I cannot be totally certain as their mothers surname is recorded as Haughey, Mattaffe and MacCaffre, and I am too hopelessly inexperienced with name variants for it to be beyond a reasonable doubt. James marries Catherine Treanor in 1868, and together thay have 10 children over 20-ish years: Mary (1869), Catherine (1871), John (1873), Matilda (1875), Bridget (1878), Alice (1881-1883), Alice (1884), James (1886), and twins Maggie and Sarah in 1889. Mary and Catherine both died of TB in 1893 and 1894 respectively, and baby Alice died of scarlatina aged two. Matilda, James, Alice, Sarah, and John are all accounted for with mother Catherine in Scotland after 1898, however Bridget, James Sr., and Maggie are missing. At least one of the girls is living as per the 1911 Scotland census, which states Catherine has had 10 children and 6 are living.

I am relatively certain this is all the same family, as Catherine Treanor lists 36 Chapel street as her residence/place of birth for most of her children and James Murray, Shoemaker as their father. John is born on High Street, and Bridget on Kilmorey Street, but she is registered by a Matilda Murray of High Street. Matilda Murray of High Street dies in 1890 aged 70, and the death is reported by a Catherine Murray, daughter in law, of 11 Chapel Street, the address listed as the birthplace of the twins. The 1901 census records a John Murray aged 45 in Chapel Street with children John, Matilda, and Patrick, so I am thinking perhaps relatives? but unfortunately anyone distinctive enough to determine a relationship is dead or is in Scotland by then. I cannot find any marriages for John Murray and Matilda Haffy/Haughey/variant of, so no hint as to who their parents may be to determine birth dates, no deaths for the elder John, Bridget, Maggie, or James Sr, and nothing on Catherine Treanor bar the name James for her father, his occupation as a Watchman, and that she may have had a sister Mary and a mother Catherine. I am also wondering if they came over during the Plantation of Ulster, or if they originate from Ireland. They are Roman Catholics, if that helps, but I am very very inexperienced with this kind of thing, and am totally reliant on records available online as I am from Australia.

I am so so very sorry to be so long winded, but I am very deep down this rabbit hole and I fear I've spent so long staring at the name Murray that I wonder if perhaps I simply I cannot see the forest for the trees, and that someone else might be able to point me in the right direction. Thank you very much in advance for any assistance anyone might be able to provide