Use TimeSeries:
Combine values and their respective times into a TimeSeries expression.
In[]:=
ts=TimeSeries[{1,3,2,3,1},{{1,1.9,2.5,3,10}}]
Out[]=
TimeSeries
Time: 1. to 10.
Data points: 5

Perform a moving map with a left-aligned window of size 1.
In[]:=
result=MovingMap[Total,ts,{1,Left},None]
Out[]=
TimeSeries
Time: 1. to 10.
Data points: 5

Display the resulting values.
In[]:=
Values[result]
Out[]=
{4,5,5,3,1}