  Let $n\geq 2$. Let $VB_n$ (resp. $VP_n$) denote the virtual braid group
(resp. virtual pure braid group), let $WB_n$ (resp. $WP_n$) denote the welded
braid group (resp. welded pure braid group) and let $UVB_n$ (resp. $UVP_n$)
denote the unrestricted virtual braid group (resp. unrestricted virtual pure
braid group). In the first part of this paper we prove that, for $n\geq 4$, the
group $VP_n$ and for $n\geq 3$ the groups $WP_n$ and $UVP_n$ are characteristic
subgroups of $VB_n$, $WB_n$ and $UVB_n$, respectively. In the second part of
the paper we show that, for $n\geq 2$, the virtual braid group $VB_n$, the
unrestricted virtual pure braid group $UVP_n$, and the unrestricted virtual
braid group $UVB_n$ have the R$_\infty$-property. As a consequence of the
technique used for few strings we also prove that, for $n=2,3,4$, the welded
braid group $WB_n$ has the R$_\infty$-property and that for $n=2$ the
corresponding pure braid groups have the R$_\infty$-property. On the other hand
for $n\geq 3$ it is unknown if the R$_\infty$-property holds or not for the
virtual pure braid group $VP_n$ and the welded pure braid group $WP_n$.
