This is incredibly hacky and hard coded. But it gives you much flexibility in changing the look to your taste. The only bad thing: Curved arrows look bad, as the bending radius does not adapt properly. Try to use small curvature.
% arara: pdflatex
\documentclass{article}
\usepackage{tikz-cd}
\tikzcdset{%
triple line/.code={\tikzset{%
double equal sign distance, % replace by double distance = 'measure'
double=\pgfkeysvalueof{/tikz/commutative diagrams/background color}}},
quadruple line/.code={\tikzset{%
double equal sign distance, % replace by double distance = 'measure'
double=\pgfkeysvalueof{/tikz/commutative diagrams/background color}}},
Rrightarrow/.code={\tikzcdset{triple line}\pgfsetarrows{tikzcd implies cap-tikzcd implies}},
RRightarrow/.code={\tikzcdset{quadruple line}\pgfsetarrows{tikzcd implies cap-tikzcd implies}}
}
\newcommand*{\tarrow}[2][]{\arrow[Rrightarrow, #1]{#2}\arrow[dash, shorten >= 0.5pt, #1]{#2}}
\newcommand*{\qarrow}[2][]{\arrow[RRightarrow, #1]{#2}\arrow[equal, double distance = 0.25pt, shorten >= 1.28pt, #1]{#2}}
\begin{document}
\begin{tikzcd}
\arrow{r}{1}\arrow{d}[swap]{1}
& \null \arrow[Rightarrow]{r}{2}\arrow[Rightarrow]{d}[swap]{2}
& \null \tarrow["3"]{r}\tarrow[swap, "3"]{d}
& \null\qarrow["4"]{r}\qarrow[swap, "4"]{d}
& \null \\
\null\arrow[bend right]{dr}
& \null\arrow[crossing over]{dl}\arrow[Rightarrow, bend right]{dr}
& \null\arrow[Rightarrow, crossing over]{dl}\tarrow[bend right]{dr}
& \null\tarrow{dl}\qarrow[bend right]{dr} % crossing over not working with this command. has to be done manually
& \null\qarrow{dl} \\ % crossing over not working with this command. has to be done manually
\null & \null & \null & \null & \null \\
\end{tikzcd}
\end{document}

If you prefer the xy
look, you can just set the shaft thickness to bigger values:
% arara: pdflatex
\documentclass{article}
\usepackage{tikz-cd}
\tikzcdset{%
triple line/.code={\tikzset{%
double distance = 3pt,
double=\pgfkeysvalueof{/tikz/commutative diagrams/background color}}},
quadruple line/.code={\tikzset{%
double distance = 5.3pt,
double=\pgfkeysvalueof{/tikz/commutative diagrams/background color}}},
Rrightarrow/.code={\tikzcdset{triple line}\pgfsetarrows{tikzcd implies cap-tikzcd implies}},
RRightarrow/.code={\tikzcdset{quadruple line}\pgfsetarrows{tikzcd implies cap-tikzcd implies}}
}
\newcommand*{\tarrow}[2][]{\arrow[Rrightarrow, #1]{#2}\arrow[dash, shorten >= 0.5pt, #1]{#2}}
\newcommand*{\qarrow}[2][]{\arrow[RRightarrow, #1]{#2}\arrow[equal, double equal sign distance, shorten >= 3.3pt, #1]{#2}}
\begin{document}
\begin{tikzcd}
\arrow{r} & \null \arrow[Rightarrow]{r} & \null \tarrow{r} & \null\qarrow{r} & \null
\end{tikzcd}
\end{document}

A compromise for the bending problem would be to define own commands for the bent arrows which have their inner line shortened by 2.2pt. This would look like this:
% arara: pdflatex
\documentclass{article}
\usepackage{tikz-cd}
\tikzcdset{%
triple line/.code={\tikzset{%
double equal sign distance, % replace by double distance = 'measure'
double=\pgfkeysvalueof{/tikz/commutative diagrams/background color}}},
quadruple line/.code={\tikzset{%
double equal sign distance, % replace by double distance = 'measure'
double=\pgfkeysvalueof{/tikz/commutative diagrams/background color}}},
Rrightarrow/.code={\tikzcdset{triple line}\pgfsetarrows{tikzcd implies cap-tikzcd implies}},
RRightarrow/.code={\tikzcdset{quadruple line}\pgfsetarrows{tikzcd implies cap-tikzcd implies}}
}
\newcommand*{\tcurvedarrow}[2][]{\arrow[Rrightarrow, #1]{#2}\arrow[dash, shorten >= 2.2pt, #1]{#2}}
\newcommand*{\qcurvedarrow}[2][]{\arrow[RRightarrow, #1]{#2}\arrow[equal, double distance = 0.25pt, shorten >= 2.2pt, #1]{#2}}
\begin{document}
\begin{tikzcd}[column sep=2]
\null\arrow[bend right]{dr}
& \null\arrow[Rightarrow, bend right]{dr}
& \null\tcurvedarrow[bend right]{dr}
& \null\qcurvedarrow[bend right]{dr}
& \null \\
& \null & \null & \null & \null \\
\end{tikzcd}
\end{document}
