arcgis desktop - Select points approx. 2000 metres from another point along a river?

I have a series of points representing a river network (See Image Above). Each point contains attributes on Latitude, Longitude, Z value and Length calculated using flow accumulation. The river network was created from Digital Elevation Models using hydrology tools in ArcGIS (Fill, Flow Direction, Flow Accumulation, Strahler Order > 4). I also have a polyline shapefile that also runs along this river network.

Starting at the first point at the beginning of each network I would like to determine another point "downstream" that is approximately 2000 metres away (Or closest to 2000 metres). I would then like to calculate channel slope based on the difference in elevation divided by the difference in length (2000m) using the two points.

I am aware that at the junctions there will be an issue, any suggestions on accomplishing this task in GIS to determine the point 2000m downstream and continuing downstream at junctions?

I am using ArcGIS 10.1 with an ArcInfo license and also have access to QGIS and GrassGIS. I am familiar with using python scripts as well.

Edit: Progress

The stream polyline has the following attributes:

x1: x coordinate of start point (Double)

y1: y coordinate of start point (Double)

x2: x coordinate of end point (Double)

y2: y coordinate of end point (Double)

length: length of polyline

shape: shape of polyline

startxy: [x1]&", "&[y1] (Text)

endxy: [x2]&", "&[y2] (Text)

I am using the tool to make arrays

start = arcpy.da.FeatureClassToNumPyArray("polyline",["x1","y1"])

Dictionary

G ={}
>>> for (st,sh,le,en) in zip (start,shape,length,end):
...     G[st]=(st,le,en)

Algorithm

def Downstream (p, x, G):
...     e = G[p]
...     if (IsNull(e)):
...         return ("Not Found")
...         if (x < 0):
...             return ("Invalid")
...             if (x < e.length):
...                 return (Along (e.shape, x))

...                 return (Downstream (e.end, x-e.length,G))

The Downstream function is receiving a runtime error where there is a key error. I have used G.keys() to see the keys, but they are not working when inputted in the function.