I am quoting from The Memoirs of Sherlock Holmes, The Reigate Squires by Arthur Conan Doyle :"The colonel waved his hand towards my friend and the inspector bowed.'We thought that perhaps you would care to step across, Mr. Holmes'"(end of quote). It is clear from the context that the inspector is asking Holmes if he wanted to intervene in the ongoing investigation, but the usage of 'to step across' confuses me, especially as I couldn't find the verb in online dictionaries. Had he used 'to step into', it would have made sense, but not 'to step across'. Well, at least to me.
The context is that Holmes and Watson are staying with the Colonel at his house when the butler comes into the room to announce that the coachman at a nearby house (Acton's house) has been murdered. The Colonel and Acton are both men of importance and the houses may therefore be imagined to be large and well separated from each other by grounds and road or drives.
“Ah, then, we’ll step over afterwards,” said the Colonel, coolly settling down to his breakfast again ...
The phrase "step over" means that they will walk to the other house. It refers to taking footsteps from one place to another, perhaps taking steps over the intervening ground or street.
The butler re-enters:
“Inspector Forrester, sir,” said the butler, throwing open the door.
The official ... stepped into the room. “Good-morning, Colonel,” said he; “I hope I don’t intrude, but we hear that Mr. Holmes of Baker Street is here.”
The Colonel waved his hand towards my friend, and the Inspector bowed.
“We thought that perhaps you would care to step across, Mr. Holmes.”
The inspector is inviting Holmes to step across from one house to the other, to walk across the intervening ground or street, similar to step over.