No reason why not, except the stair nosing may not fit exactly right as there could be a slight variation in the thickness or groove profile. Easy to fix with a table saw/shim for the most part. Have to see it.
I learned this business from an out fit that never supplied prefinished manufacturer moldings. All but the stair nosing and quarter round were either finished from their own unfinished stock or you milled your own (table saw) and had the shop guy, sand, stain and put a matching finish sheen on it.
By way of milling your own, this was done with the supplied hardwood (if it was a 3/4" hardwood) for the job or we would just grab some unfinished leftover 3/4" material from warehouse stock.
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So, do you think it is okay of I buy cheaper transition pieces from someone other than Mirage? Are the transition pieces engineered or solid?
Should be solid, but nowadays one never knows what one manufacturer to the other may have. Most should be solid. I recall Kahrs at one time didn't have solid trim moldings for a few rare species of hardwood. Unknown why. Perhaps they didn't or could not buy thicker stock material in that species to make the moldings. Instead they were laminated.