rationalskepticism.org

The meaning usually stays the same. For example, when the character 年 was introduced to Japan (with the meaning 'year' and a reading based on the contemporary Chinese pronunciation), the Japanese language already had their own word for 'year' ("toshi"). As the writing system developed to also cover Japanese words, the same character 年 was used also for the Japanese native word. So both "nen" and "toshi" mean the same thing ('year'). Which reading to use for the character just depends on the context and the history of the word or compound in question.

It's sort of similar how English has a layer of latinate words alongside native words--had Latin used a logographic writing system, in today's English there very well might exist a character for teeth that was read either "dent-", "tooth" or "teeth".

Edit: didn't realise the better example would be the above 年 having the readings "annu-" and "year".