I'm the midst of a history-inspired collaborative fiction (roleplaying) game in which a North Carolina wilderness confrontation results in the death of wealthy aristocrat who was a Confderate colonel just a few years back. It wasn't exactly cold-blooded murder, but neither was it a fair standoff -- he died with a bible in his hand, not a gun. The shooters trek overland and out of state. The victim's relatives witnessed the act and carry the body back into Raleigh, and would tell the story to authorities in the most onerous light.
In the 21st century, DNA would be collected, facial recognition software employed, FBI databases updated, alerts sent out to local PDs, etc.
What would have happened in 1875ish?
First thing is to get a sketch and a report of anything noteworthy (gun type, description of horses if any, description of what the men were carrying, description of what the men were wearing, most especially everything you can remember about what the men said, names, places, type of work.
How far away are the men expected to be? If they are still within a weeks travel at about 20-30 miles per day you can try to round up a posse and go after them. Get a judge to sign an arrest warrant. Ask the Sheriff for help.
If the men are beyond reach you send out word on the telegraph to the most likely places they will go. Promise a reward and you've got a better chance at information. If they likely went somewhere without telegraphs you can send riders out with the sketch and description. Post fliers up wherever you can. Use the post office to send advertisements to as many newspapers as you can afford.
If you have money you could hire a detective agency like the Pinkerton Detective Agency. A Private Detective would thoroughly examine your evidence and likely take some of the above described steps. Plus, as long as you were paying he or they would hunt for the men responsible. The arrest warrant is important. The Pinkertons considered themselves respectable and within the law. They weren't a private assassin group. They were open to questionable acts but they weren't going to just kill somebody on your word.
Your last option is vigilante justice. You can go after the man yourself without legal help. It makes for a great movie. However, your odds of finding the man are better with the help of professionals.
A lot of police work really hasn't changed in the past 140 years - a local sheriff may interview people who had possible motives, and the local press, particularly if the victim was wealthy enough, would run gruesome stories detailing the murder (along with shocking pictures - the press had less scruples then, even running photographs and names of rape victims). If the victim was really quite prominent, the local press would ride that story for a long time to sell papers.
If, after a substantial amount of searching, the local sheriff couldn't find the party(-ies) responsible, then you'd likely be out of luck. Case clearance wasn't very high back then, I'd suspect.
But I think a lot of violent crime back then - like now - would have been committed by persons with grudges or with opportunity, both of which are fairly well traceable. You'll see a lot of old news items - up to and including the 1930's in the mountains of North Carolina - that detail just absolute hearsay and rumor used by the police to find someone with a motive.
Of course, if this was the other way around, and it was a former Union colonel killed in front of his family, then the local sheriff might mysteriously be unable to find any evidence to even identify a single suspect...
Sources: James Rusher, Until He is Dead: Capital Punishment in Western North Carolina; Jim Wise, Murder in the Courthouse: Reconstruction and Redemption in the North Carolina Piedmont.