What is the origin of the idiom "change one's horse midstream" and when to use it?

The origin of the idiom "change one's horse midstream" can be traced back to an anecdote attributed to Abraham Lincoln. While the exact historical veracity of the story is debated, it is commonly believed that Lincoln used this metaphor during his presidential campaign in 1864. According to the anecdote, Lincoln was approached by a group of supporters who urged him to switch his vice presidential running mate. In response, Lincoln allegedly used the analogy of not changing horses midstream to convey his reluctance to change course or make significant changes during a critical phase of a process.