List of Figures & Tables - how to move caption label into caption itself

Question

I'm using the article document class (as a base) and the caption package. I essentially need to change my LoF format from the default

List of Figures  
1 Multiline caption multiline caption multiline caption multiline caption multiline caption
  multiline caption multiline caption
2 Multiline caption multiline caption multiline caption multiline caption multiline caption
  multiline caption multiline caption

to the following:

List of Figures
Figure 1. Multiline caption multiline caption multiline caption multiline caption multiline
caption multiline caption multiline caption
Figure 2. Multiline caption multiline caption multiline caption multiline caption multiline
caption multiline caption multiline caption

I also need to do the same for my LoT format.

By label I mean the numbers indicating the index of the figure/table.

I'm assuming it's not possible to get the desired output while still using the labels, so how would I move the labels into their captions, remove the labels themselves, and remove the indent? (If it is possible to "inline" the label with its caption then that's all I need.)

If it's not possible to modify \listoffigures and \listoftables to do so, how would I create the needed environments to do so?

Edit: My revised version of the Peter Wilson's answer using tocloft:

\renewcommand{\@tocrmarg}{2.5pc plus1fil} % Set ToC/LoF/LoT entries to ragged-right
% Set formatting for the LoF ande LoT (fig/tab numbers >999 will clip caption)
\newlength{\mylen}
% Set formatting for LoF
\renewcommand{\cftfigpresnum}{Figure } % goes before figure number
\renewcommand{\cftfigaftersnum}{.} % Goes after figure number
\settowidth{\mylen}{\cftfigpresnum} % space required to print \cftfigpresnum
\addtolength{\mylen}{\cftfignumwidth} % plus space for the number
\renewcommand{\cftfigaftersnumb}{\hspace{\mylen}} % add space after the zero-spaced number
\cftsetindents{figure}{0pc}{0pc} % Remove caption indent, and flush entries with title
\setlength{\cftbeforefigskip}{0.5pc} % Add small gap between LoF entries
% Set formatting for LoT
\renewcommand{\cfttabpresnum}{Table }
\renewcommand{\cfttabaftersnum}{.}
\settowidth{\mylen}{\cfttabpresnum}
\addtolength{\mylen}{\cfttabnumwidth}
\renewcommand{\cfttabaftersnumb}{\hspace{\mylen}}
\cftsetindents{table}{0pc}{0pc}
\setlength{\cftbeforetabskip}{0.5pc}

Edit2: How do I move the label or flush the caption left with tocbasic instead of tocloft? MWE:

\documentclass{article}
%\documentclass[listof=totoc,listof=leveldown]{scrartcl}
\usepackage{lipsum}
\usepackage{hyperref}% Used in actual project, provides \phantomsection

\usepackage[justification=raggedright,singlelinecheck=false]{caption}
\usepackage{float}
\usepackage{pgffor}
\usepackage{tocbasic}
%\tocbasicautomode % What does this do?
\newcommand*{\addfigdot}[1]{\figurename\ #1\unskip.\hfil}
\newcommand*{\addtabdot}[1]{\tablename\ #1\unskip.\hfil}
\DeclareTOCStyleEntries[
    %breakafternumber,% Somehow manages to flush at least the first line left?
    beforeskip=.5pc,% Add space between entries
    indent=0pc,% Remove basic indent
    entrynumberformat=\addfigdot,% Reformat prefix to "Figure <num>. "
    raggedentrytext,% Disable caption justification in LoF
    dynnumwidth% Needed to properly align caption with prefix
]{tocline}{figure,table}
\DeclareTOCStyleEntry[entrynumberformat=\addtabdot]{tocline}{table}

\begin{document}
\tableofcontents

\section{Figures and Tables}
\foreach\n in {1,...,3}{\begin{figure}[H]\caption{\lipsum[\n][1-3]}\end{figure}}
\foreach\n in {1,...,3}{\begin{table}[H]\caption{\lipsum[\n][1-3]}\end{table}}

% Print LoF and LoT, and add them to ToC
\cleardoublepage
\phantomsection\addcontentsline{toc}{section}{\listfigurename}
\listoffigures
\cleardoublepage
\phantomsection\addcontentsline{toc}{section}{\listtablename}
\listoftables
\end{document}

Answer 1

Try this (less any typos):

\documentclass{article}
\usepackage{tocloft}
\newlength{\mylen} % a scratch length
\renewcommand{\cftfigpresnum}{Figure } % goes before figure number
\settowidth{\mylen}{\cftfigpresnum} % space required to print \cftfigpresnum
\addtolength{\mylen}{\cftfignumwidth} % plus space for the number
\setlength{\cftfignumwidth}{0em} % but make this zero 
\renewcommand{\cftfigaftersnumb}{\hspace{\mylen}} % add space after the zero-spaced number
\renewcommand{\cfttabpresnum}{Table } % repeat above for Tables
\settowidth{\mylen}{\cfttabpresnum}
\addtolength{\mylen}{\cfttabnumwidth}
\setlength{\cfttabnumwidth}{0em}
\renewcommand{\cfttabaftersnumb}{\hspace{\mylen}}

\begin{document}
\section{A section}
Some text

\begin{figure}
\centering
A FIGURE
\caption{A very long caption for a figure where the caption will extend over one
         line in the LoF hopefully like this}
\end{figure}
\begin{table}
\centering
A TABLE
\caption{Table caption}
\end{table}

\end{document}

All the fiddling about with lengths is to make the LoF and LoT multiline entries print with the second and later lines aligned with (in your words) the label.

Read the tocloft documentation (> texdoc tocloft) for more information.

Answer 2

A wrapper command to insert the string into \caption's short-form option can be used:

\documentclass{article}

\usepackage{lipsum}
\usepackage{float}
\usepackage{etoolbox}
\usepackage[listformat=empty]{caption} % Remove figure/table numbers from LoF/LoT
\usepackage{tocloft}
    % Add spacing between LoF/LoT entries
    \setlength{\cftbeforefigskip}{0.5pc}
    \setlength{\cftbeforetabskip}{0.5pc}
    % Remove LoF/LoT indents
    \cftsetindents{figure}{0pc}{0pc}
    \cftsetindents{table}{0pc}{0pc}

\newcommand\makecaptioncmd[1]{%
    \newcommand\mycaption[1]{{\caption[#1 {##1}]{{##1}}}}}
\AtBeginEnvironment{figure}{%
    \makecaptioncmd{\figurename\ \thefigure.}}
\AtBeginEnvironment{table}{%
    \makecaptioncmd{\tablename\ \thetable.}}

\begin{document}
\begin{figure}[H]\mycaption{\lipsum[1][1-3]}\end{figure}
\begin{figure}[H]\mycaption{\lipsum[2][1-3]}\end{figure}
\begin{table}[H]\mycaption{\lipsum[3][1-3]}\end{table}
\begin{table}[H]\mycaption{\lipsum[4][1-3]}\end{table}
\listoffigures
\listoftables
\end{document}