How to reproduce the bug:
1 - Create new project using UE 4.16.1 (I used Desktop/Console With Starter Content).
2 - Create a new level and in this, add an existing level (I added the Minimal_Default level). The bug seems to be the same for both Streaming Method (Blueprint or Always Loaded).
3 - Create a Level Sequence (is not important if is or not in the Persistent Level, the bug is the same).
4 - Add an StaticMeshActor in the Persistent Level and another one in the second level. Animate those StaticMeshActors in the new created Level Sequence.
To play the sequence, I added in the Level Blueprint an event when a key is pressed (the bug is the same if I used the Level Blueprint from Persistent Level or the second level or an Actor Blueprint).
If I choose to play in Standalone Game, only the StaticMeshActor from Persistent Level will be animated, but it will work perfect if I play in Selected Viewport or in New Editor Window.
This behavior seams to be valabile only for Level Sequences created in the UE 4.6.1.
I made the same think in UE 4.15.3. Once opened in 4.16.1 the sequence it works perfect even if I play in Standalone Game.