I hope a solution with animation nodes (is a free and amazing add-on and still can do a lot more than geometry nodes) is ok, so with this node setup:

with this node:

you can change the speed of the "follower".

you get this (slow speed):

moderate speed:

fast:

I use the compare node to reset the animation. In combination with the switch node AN sets the start point for the cone to 0,-1,0 so that it restarts always at the same point. This only happens on frame 1, on every other frame the "normal" "follow object calculation" will be executed.

You can try out yourself what happens if you just plug in the vector math output node of "add" to the location of "object transform output". You will see, that the starting point of cone is "on" the cube (at least if you have set execution "always" in the node tree). That's because AN executes the nodetree permanently and not per frame. Of course you can change that behaviour.

I hope a solution with animation nodes (is a free and amazing add-on and still can do a lot more than geometry nodes) is ok, so with this node setup:

with this node:

you can change the speed of the "follower".

you get this (slow speed):

moderate speed:

fast:

I use the compare node to reset the animation. In combination with the switch node AN sets the start point for the cone to 0,-1,0 so that it restarts always at the same point. This only happens on frame 1, on every other frame the "normal" "follow object calculation" will be executed.

You can try out yourself what happens if you just plug in the vector math output node of "add" to the location of "object transform output". You will see, that the starting point of cone is "on" the cube (at least if you have set execution "always" in the node tree). That's because AN executes the nodetree permanently and not per frame. Of course you can change that behaviour.

Now to "your" blend file:

i am not totally sure why this happens. But...that's why i moved my object on the curve via AN too, then it works.

Just delete your follow path constraint and use the same nodes i did: Time info, map range, evaluate spline and transform output. These four nodes replace the follow path constraint.

I hope a solution with animation nodes (is a free and amazing add-on and still can do a lot more than geometry nodes) is ok, so with this node setup:

with this node:

you can change the speed of the "follower".

you get this (slow speed):

moderate speed:

fast:

I hope a solution with animation nodes (is a free and amazing add-on and still can do a lot more than geometry nodes) is ok, so with this node setup:

with this node:

you can change the speed of the "follower".

you get this (slow speed):

moderate speed:

fast:

I use the compare node to reset the animation. In combination with the switch node AN sets the start point for the cone to 0,-1,0 so that it restarts always at the same point. This only happens on frame 1, on every other frame the "normal" "follow object calculation" will be executed.

You can try out yourself what happens if you just plug in the vector math output node of "add" to the location of "object transform output". You will see, that the starting point of cone is "on" the cube (at least if you have set execution "always" in the node tree). That's because AN executes the nodetree permanently and not per frame. Of course you can change that behaviour.