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We derive an analytical approximation for the linear scaling evolution of the characteristic length <i>L</i> and the root-mean-squared velocity <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>σ</mi><mi>v</mi></msub></semantics></math></inline-formula> of standard frictionless domain wall networks in Friedmann–Lemaître–Robertson–Walker universes with a power law evolution of the scale factor <i>a</i> with the cosmic time <i>t</i> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>∝</mo><msup><mi>t</mi><mi>λ</mi></msup></mrow></semantics></math></inline-formula>). This approximation, obtained using a recently proposed parameter-free velocity-dependent one-scale model for domain walls, reproduces well the model predictions for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> close to unity, becoming exact in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>→</mo><msup><mn>1</mn><mo>−</mo></msup></mrow></semantics></math></inline-formula> limit. We use this approximation, in combination with the exact results found for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, to obtain a fit to the model predictions valid for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula> with a maximum error of the order of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>%</mo></mrow></semantics></math></inline-formula>. This fit is also in good agreement with the results of field theory numerical simulations, especially for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>∈</mo><mo>[</mo><mn>0.9</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>. Finally, we explicitly show that the phenomenological energy-loss parameter of the original velocity-dependent one-scale model for domain walls vanishes in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>→</mo><msup><mn>1</mn><mo>−</mo></msup></mrow></semantics></math></inline-formula> limit and discuss the implications of this result.