System Information
Operating system: Windows-10-10.0.19041-SP0 64 Bits
Graphics card: AMD Radeon RX 5700 XT ATI Technologies Inc. 4.5.14757 Core Profile Context 20.12.1 27.20.14501.28009

Blender Version
Broken: version: 2.93.3, branch: master, commit date: 2021-08-17 18:30, hash: 8b80d19f3641, type: release
Worked: n/a

Short description of error
When using keyed between two objects, the render output is something else than is visible on the viewport.

Exact steps for others to reproduce the error
[Please describe the exact steps needed to reproduce the issue]

1. Start Blender
2. Remove square
3. Add UV sphere
   1. Move it:
      • g / y / -8
4. Add plane
5. Add particle
   • Emission: Start 5
   • Emission: End 85
   • Velocity: Normal 0,5 m/s
   • Render: Render as object
     • Instance Object: Plane
   • Field Weights: Gravity: 0
6. Duplicate sphere to Sphare.001 (Ctrl+d)
   1. Move it:
      • g / y / 16
7. On the particular
   1. Make a single-user copy (by pressing [2] on the settings section)
      • Emission: Start 45
      • Emission: End 120
      • Physics: Physics Type: Keyed
      • Relations:
        • Sphere: particle-system
        • Sphere.001: particle-system
8. Move the camera to see both balls
9. Go to frame 71
10. Render the frame
11. Done

On the viewport I have the following:

And the rendered output looks like following:

Comment

I already reported on the previous version:
Case T87782: Keyed rendering differs from the viewport
But I was not quick enough :)

I still believe, this is so called low-hanging fruit as on the viewport this looks correct and expected. But only when doing the rendering, this problem appears.

Edit:
Baking the dynamics is tested as it was proposed earlier: no help.

best regards
Petri