How to show visual routing and virtually driving the route on an OSM-based map in my web app?

Hello all,

I'm working on a web based application which works around the OSM Map. From a database, 'jobs' are generated at a coordinate and a map marker is created. There are vehicles (virtual, not real) also which are on the map as markers.

What I want to achieve is a way of getting the vehicles (so the marker) to drive to a job if directed to by the user. Effectively having the marker move, following a route. The route itself doesn't have to be visible but if it can that would be a plus. I have a backend way of linking vehicle to the job and it knowing it needs to travel there, but I have not yet worked out a way to visually route this.

The application is not-for-profit but could face quite high demand on any routing service (potentially 30-40 jobs and 100+ vehicles all 'routing' at any one time), so I am looking for a free or cheap option as best as possible, although I do appreciate the demands my application has, it may mean I don't have the option of a free/cheap service.

Additionally I need a way of getting my list of vehicles and quickly checking which unit is possibly the closest by drive time, but not essential if my options are limited in my original requirement.

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I am asking if users of OSM know of any available methods, be it plugins or otherwise that will allow me to move a marker along a path, generated from a route. In other words, simulating travel. I'm aware of various routing systems out there but unsure if choosing a particular one assists or prevents me in completing my simulated travel.

Hope that all makes sense, appreciate the help.

Regards, Jason