I am into web development and I want to check my web page for different browsers.
So I am searching for an offline tool to check cross-browser compatibility.
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You can fire up a virtual machine with a specific browser installed and test against that.
There are various possible sources of the VM images:
You might also like to take a look at the Android Developers SDK for Android Studio as this includes emulators for various Android Devices with different versions of Android, screen sizes, etc., and you can potentially download optional browsers to the emulators. I suspect that the SDKs for other mobile platforms will provide something similar.
You would also be well advised to check your code with one or more of the W3C validators here, they are available online but most, if not all, of them have the code available to download and run yourself offline - click on check the code and look for readme files for instructions.