How to export FBX at runtime with animation(s), rig and the mesh to Maya, blender etc.. ?

Here's a working code snippet that should at least be a good starting point:

 private static void ExportFbxScene(string filename) {
     var manager = FbxManager.Create();
     manager.SetIOSettings(FbxIOSettings.Create(manager, Globals.IOSROOT));
     var exporter = FbxExporter.Create(manager, "exporter");
     exporter.Initialize(filename, /*FileFormat=*/-1, manager.GetIOSettings());
     var scene = FbxScene.Create(manager, "scene");
     AddSkeletonTo(ref scene);
     exporter.Export(scene);
 }
 
 private static void AddSkeletonTo(ref FbxScene scene) {
     var rootObj = FbxNode.Create(scene, "root-obj");
     var bones = new List<FbxNode>();
     var boneNames = new[]  { "root-bone", "child-bone" };
     scene.GetRootNode().AddChild(rootObj);
 
     for (var i = 0; i < boneNames.Length; i++) {
         // create the bone object
         var skeletonAttribute = FbxSkeleton.Create(scene, boneNames[i]);
         skeletonAttribute.SetSkeletonType(i == 0 ?
             FbxSkeleton.EType.eRoot : FbxSkeleton.EType.eLimbNode);
         var bone = FbxNode.Create(scene, boneNames[i]);
         bone.SetNodeAttribute(skeletonAttribute);
         // set the bone's transform
         bone.LclTranslation.Set(new FbxDouble3(0, -5, 0)); // x, -y,  z
         bone.LclRotation.Set(new FbxDouble3(0, -0, -0));   // x, -y, -z
         bone.LclScaling.Set(new FbxDouble3(1, 1, 1));      // x,  y,  z
         // add to parent bone
         if (i > 0) bones[i - 1].AddChild(bone); // link: A -> B -> C -> ...
         // add to local bone list for next iteration
         bones.Add(bone);
     }
     
     var pose = FbxPose.Create(scene, boneNames[0]);
     pose.SetIsBindPose(true); // default is rest pose
     foreach (var bone in boneNodes) {
         var bindMatrix = bone.EvaluateGlobalTransform();
         pose.Add(bone, new FbxMatrix(bindMatrix));
     }
     scene.AddPose(pose);
 }

Sources I used:

• Unity FBX Developer's Guide

• Import FBX Animations Using C

• Autodesk FBX SDK API Reference

• Snippets from various online repos