You send the entire history for that conversation every time and likely more if its getting info from tools. If its not in the context the model dose not see it unless you have a memory system that dose something like feeding in summaries of past conversations that also takes up tokens and context. Rolling drops old messages to not reach context limits but you can lose important info or get odd results. If the history gets bigger than the context things break or slow way down.

Most agent harnesses do something called "compaction." For example, here's how Pi does compaction

does it give the full history to the LLM each time?

It's limited to the context size supported by given model. You can give the model 100k tokens of history but if it's configured for less, it will just truncate it before processing (usually by removing oldest tokens first)