QGIS Assign point data to a line layer considering line direction

There is a possibility using a "Virtual Layer" through Layer > Add Layer > Add/Edit Virtual Layer...

Let's assume there are five features in 'rivers' (for simplification "id" is equal to "Zone") and five features in 'some_points' layer accordingly, where points are located at the start of each 'rivers''s feature, see image below.

With the following query, it is possible to assign the data associated with each point to the line segment following it, when considering the line direction.

SELECT r.*, sp.DataField
FROM rivers AS r, some_points AS sp 
WHERE st_intersects(StartPoint(r.geometry),sp.geometry)

The output Virtual Layer will maintain initial attributes and geometries of 'rivers' and additional field "DataField" from 'some_points' layer.

Here's another approach using the QGIS Field Calculator (if SQL and Virtual Layers are not your things - although personally, I'm a big fan).

Add a calculated field to the 'rivers' layer:

aggregate(layer:='some_points',
          aggregate:='max',
          expression:="DataField",
          filter:=intersects($geometry, start_point(geometry(@parent)))
          )

This will retrieve the maximum value (there should be only one anyway) of "DataField" from the 'some_points' layer where the point intersects the first point of the 'rivers' geometries.

This will only work in QGIS 3. If you want multiple values from the 'some_points' layer you would need to add a calculated column for each one.

If you want lots of columns or need to support QGIS 2 the virtual layers from @Taras is the way to go.