"追赶入洞穴" can be translated into English as "chase (someone/something) into the cave" or "pursue (someone/something) into the cave", depending on the context and the degree of formality you prefer.
"Chase" is a more common and casual term.
"Pursue" is slightly more formal and can imply a more determined or persistent effort.
You can choose either based on your specific needs. For example:
They chased the thief into the cave.
The hunters pursued the wild animal into the cave.