Can I programmatically move the steps of a mat-horizontal-stepper in Angular / Angular Material

Yes. It is possible to jump to a specific stepper by using selectedIndex property of the MatStepper. Also, MatStepper exposes public methods next() and previous(). You can use them to move back and forth.

In your template:

Add an id to your stepper e.g. #stepper. Then in your goBack() and goForward() methods, pass the stepper id:

<mat-horizontal-stepper #stepper>
    <!-- Steps -->
</mat-horizontal-stepper>    
<!-- second option -->
<div>
   <button (click)="goBack(stepper)" type="button">Back</button>
   <button (click)="goForward(stepper)" type="button">Next</button>
</div>

.. and in your typescript:

import { MatStepper } from '@angular/material/stepper';

goBack(stepper: MatStepper){
    stepper.previous();
}

goForward(stepper: MatStepper){
    stepper.next();
}

Link to stackblitz demo.

You can also use ViewChild to get a reference to the stepper component in your TypeScript as shown below:

@ViewChild('stepper') private myStepper: MatStepper;

goBack(){
    this.myStepper.previous();
}

goForward(){
    this.myStepper.next();
}

In this case, you don’t have to pass the stepper reference in the method in your component’s html. Link to Demo with ViewChild

You can enable/disable the Back and Next buttons by using the following:

<button (click)="goBack(stepper)" type="button" 
        [disabled]="stepper.selectedIndex === 0">Back</button>
<button (click)="goForward(stepper)" type="button" 
        [disabled]="stepper.selectedIndex === stepper._steps.length-1">Next</button>