The answer is, it depends on how many shards (and replicas) your index(es) have.
Every index is divided into 1 or more shards. By default I think you get 5 shards.
The shard count cannot be changed for an existing index.
Each shard has by definition one primary copy. It can (should) also have 1 to N replicas (think, redundant copies).
It is typical to maintain 1 or 2 replicas (or in some cases more) of every shard. Then when a shard is lost (node goes offline, maybe forever), service is not interrupted. New replicas are automatically remade.
The replica count can be changed for an existing index. It can be set low during initial indexing, then increased once indexing is complete, for example.
To answer your question,
By default, ES will attempt to balance all the shards (primaries and their replicas) across the available data nodes. Unless disabled, shards will autobalance according to reasonably conservative and appropriate defaults.
So: unless your index was built with only one shard, one would expect shards to migrate when a new data node joins the cluster.
(Data node means: a node that is allowed to hold shard data. In large cluster typically master nodes do not hold data. It is also possible to have client nodes, which handle queries but do not hold data or act as masters. More types are coming...)
There are a lot of settings related to this you will want to understand and tune appropriately for your index and use case!