Increase spacing after minipage

Below there's one possible solution; the key ideas were:

1. I used the solution given in How to keep a constant baselineskip when using minipages (or \parboxes)? to guarantee spacing after the minipages.

2. Before the minipages I used \smallskip\nointerlinespacing.

3. I defined a newenvironment sminipage (to apply \small inside minipage); the optional argument (set by default to t) controls the alignment of the minipage and the mandatory argument sets the width of the minipage.

I also made some other modifications to improve your code:

4. I removed the spurious blank space after the first \end{minipage}.

5. I defined a \Pder command to facilitate the writing of the partial derivatives.

The idea was to obtain (approximately) the same spacing around the equations inside the minipage as the one used for regular equations not inside a minipage (I added a regular equation environment at the end just for comparison purposes):

    \documentclass{article}
    \usepackage{amsmath}
    \usepackage[nopar]{lipsum}% just to generate text for the example

    \newcommand\Pder[2]{%
      \frac{\mathrm{d}#1}{\mathrm{d}#2}}

    \newenvironment{sminipage}[2][t]
      {\minipage[t]{#2}\small}
      {\endminipage}

    \begin{document}

    \lipsum*[4]\par\smallskip\nointerlineskip
    \noindent\begin{sminipage}[t]{0.5\textwidth}
    \begin{equation}\label{e3}
    \begin{split}
    \Pder{x'}{t} &= \Pder{x}{t} - \Pder{\phantom{x}}{t} (ut) \\
    \Pder{x'}{t} &= \Pder{x}{t} - u\,\Pder{\phantom{x}}{t} (t) \\
    \Pder{x'}{t} &= \Pder{x}{t} - u\,\Pder{t}{t} \\
    v' &= v - u
    \end{split}
    \end{equation}
    \end{sminipage}%
    \begin{sminipage}{0.5\textwidth}
    \begin{equation}\label{e4}
    \begin{split}
    \Pder{x}{t} &= \Pder{x'}{t} - \Pder{\phantom{x}}{t} (ut) \\
    \Pder{x}{t} &= \Pder{x'}{t} - u\,\Pder{\phantom{x}}{t} (t) \\
    \Pder{x}{t} &= \Pder{x'}{t} - u\,\Pder{t}{t} \\
    v &= v' + u
    \end{split}
    \end{equation}\null
    \par\xdef\tpd{\the\prevdepth}
    \end{sminipage}

    \prevdepth\tpd
    \noindent\lipsum[2]
    \begin{equation}
    a=b
    \end{equation}
    \lipsum[4]

    \end{document}

![enter image description here][1]

I defined now (as was requested in a comment) a new environment eqmpage which basically is a top aligned minipage with constant width of \linewidth which automates all the preparations mentioned above:

\documentclass{article}
\usepackage{amsmath}
\usepackage[nopar]{lipsum}% just to generate text for the example

\newcommand\Pder[2]{%
  \frac{\mathrm{d}#1}{\mathrm{d}#2}}

\newenvironment{sminipage}[2][t]
  {\minipage[t]{#2}\small}
  {\endminipage}

\newenvironment{eqmpage}
  {\par\smallskip\nointerlineskip%
    \noindent\minipage[t]{\textwidth}}
  {\par\xdef\tpd{\the\prevdepth}\endminipage\par\prevdepth\tpd}

\begin{document}

\lipsum*[4]

\begin{eqmpage}
\begin{sminipage}[t]{0.5\textwidth}
\begin{equation}\label{e3}
\begin{split}
\Pder{x'}{t} &= \Pder{x}{t} - \Pder{\phantom{x}}{t} (ut) \\
\Pder{x'}{t} &= \Pder{x}{t} - u\,\Pder{\phantom{x}}{t} (t) \\
\Pder{x'}{t} &= \Pder{x}{t} - u\,\Pder{t}{t} \\
v' &= v - u
\end{split}
\end{equation}
\end{sminipage}%
\begin{sminipage}{0.5\textwidth}
\begin{equation}\label{e4}
\begin{split}
\Pder{x}{t} &= \Pder{x'}{t} - \Pder{\phantom{x}}{t} (ut) \\
\Pder{x}{t} &= \Pder{x'}{t} - u\,\Pder{\phantom{x}}{t} (t) \\
\Pder{x}{t} &= \Pder{x'}{t} - u\,\Pder{t}{t} \\
v &= v' + u
\end{split}
\end{equation}\null
\end{sminipage}
\end{eqmpage}
\noindent\lipsum[2]
\begin{equation}
a=b
\end{equation}
\lipsum[4]

\end{document}

You can make use of the command \bigskip or \medskip.

If you will always use the same size, you can define

\def\bmp{\begin{minipage}{0.48\linewidth}\small} 
\def\emp{\end{minipage}\smallskip}

for begin and end. Note the use of 48% for the width. You can change. You just need to put some \hfill between them.

\bmp \emp
\hfill
\bmp \emp
% this paragraph is important

text here....

I have upvoted @Sigur's answer. But you might also want to look at genmpage package. It adds additional options to your minipage environment and you can also write the options in the preamble.

To set your minipage text to small, put the following into your preamble:

\usepackage{genmpage}
\setkeys{GenMP}{resetfont,fsize=small,inner=s}

Then write your minipage environment as you usually do and you can use \medskip and \bigskip as suggested by @Sigur.