There have two ways to export animation.
XML Animation file
Select all needed bones, go to Main menu > Animation > Save animation > type filename, pick place, press "Save motion" button. As a result you will get animation file with extension .xaf, this is "XML Animation file". For import animation go to Main menu > Animation > Load animation > choose file with extension .xaf.
This way nobody uses because it does not convenient.
FBX file format
For export animation you need export a bones. Export them to FBX file format. When you import saved FBX file, you get moving/animated bones.
Select all needed bones, go to Main menu > File > Export > Export selected > Type filename, pick place, choose "FBX" format, press "Save" button. In "FBX export parameters" window in "Animation" list check "Animation" check box, in "Bake Animation" list check "Bake Animation" check box, set up start, end and step animation parameters, check "Resample all" check box, press "Ok" button.
This way use everybody because it does convenient.
- Before you try to export make sure you are not displaying any bones as a box. Select the object/bone, right-click (contex menu) and go to "Object Properties" group. Function "Display as Box" must be turn off, because it causes an FBX export error.
- In "FBX Settings", you must check "Convert deforming dummies to bones" option. Otherwise it will cause an error and your bones will display as geometry.
- Don't use points or helpers as bones. It is not supported and causes missing animated geometry. But if you can't avoid it then you will have to export it "twice". For some reason the "Convert deforming dummies to bones" option at export doesn't work. So first export it to FBX as you would normally do. Then import it back to 3ds Max for converting every point and helper to bones and export it out again to FBX format.
- The first thing to do is to ensure that "Show warning manager" option is checked in "UI" dropdown so that You will be aware if something went wrong during the export. For example you will see in this log if something is not supported in FBX export and is not (or partially) exported.
- You need to check what is globally used in your 3ds Max scene and what is actually animating your model. 3ds Max has some features and tools to animate models and skeletons, that are not supported in FBX format. It means that if you export your scene as FBX, these modifiers and their effects will be lost in the ouput FBX. To avoid this, you may need to use (in some cases) the "Bake animation" option in FBX Export settings. The goal of this option is to convert almost all animation to simple keyframes animation so that everything can be packed in FBX and read by other plugins.
- Ensure that "Animation", "Skin" and "Morph" exports are enabled in "Deformers" dropdown.
- You might see that baking makes your model vibrate during animation. The cause is not clear, but it comes from the 3ds Max baking function. If you experience this issue, try to change the "Step" parameter in "Bake animation" for something higher than 1 value. To give you an example, using 10 instead of 1 should be good.