World line coordinate finiteness If you trace two particles' world line backwards in time, according to current theory, both objects should converge at big bang.

Would both objects arrive there simultaneously?

Another way of asking the question:

Is there any evidence that supports the idea that new matter is "introduced" into the universe as a continuous stream, as opposed to everything coming from a singularity both in time and space?

The analogy would be from the inside of a black hole. For each particle in that singularity, they would all have a world line that starts at the same spacial coordinate, but the particles would not be "introduced" into the black hole simultaneously. You can argue that time stops having meaning in a black hole, but that's only true for an "outer observer" in my reasoning.
Particles are generated and destroyed all the time. This is obvious for photons, but also holds for massive particles. Today most particles are stable and long-living, but shortly after the big bang the universe was so hot that particle generation and destruction were in some equlibrium and all particles participated in this process (for photons this continued until the epoch of re-ionisation at redshift $z\sim1000$. There is definitely no particle around today that existed all the time since $t=0$.