Stars are born from Giant Molecular Clouds (GMCs), in big "batches" that become -- upon the condensation & dispersal of the parent GMC -- Open Clusters. Now, "stars inside an Open Cluster are at first tighly packed, moving at the same speed around the center of the Galaxy"*. Such Open Clusters typically persist for several hundred million years, before becoming disrupted**.
Now, the Ursa Major Cluster is an aged (~500 million years old*), and partially disrupted Open Cluster.
After half a billion years or so, a classic Open Cluster such as the Pleiades or the Hyades (both in Taurus) tends to be disturbed by external factors (such as Molecular Clouds passing by), setting its stars moving at slightly different speeds, and so causing them to drift apart, exactly like the one in Ursa Major has done.

When this happens, the [Open] Cluster becomes a stream of stars, not close enough to be a cluster but all related and moving in very similar directions at similar speeds...

Sirius is a former member of this [Open] Cluster and our sun is in the outskirts of what is called The Ursa Major Stream, a group of stars that are all ex-members of the Ursa Major Cluster spanning over a thousand light years in space**.
These partially disrupted post-Open Clusters are called Moving Groups, and can be up to 2 billion years old***. But, typically,
After a billion or so years, the Cluster is totally lost. Some stars will be on the far side of the galaxy, some on the near. The sun's original cluster is like this, there is no way to tell which are former members and which just happened to have formed at the same time but somewhere else.
Indeed, "extremely dense Open Clusters can stay together for longer, but no open cluster could stay together for the age of the Sun, five billion years**".