<p>The (<i>k</i>,&#xa0;<i>r</i>)-Quicksort on the fly is a sorting algorithm, which provides successively first the smallest, then the second smallest, and so on of a given set of size <i>n</i>. Let <i>X</i>(<i>n</i>,&#xa0;<i>l</i>) be the number of comparisons (respectively the time) up to the <i>l</i>th smallest shown. The correctly normalized process <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(Y_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Y</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> provides in the limit a cadlag process <i>Y</i>. The process <i>Y</i> is characterized as a stochastic fixed point. For suitable versions, the convergence <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(Y_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Y</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> to <i>Y</i> is in Skorokhod metric a.e. On the way, we also provide the average number of comparisons for the (<i>k</i>,&#xa0;<i>r</i>)-Quicksort algorithm.</p><p>The method of proof might be interesting in itself. It uses the contraction method and weighted branching processes.</p>