Carrier taking moggy across river tail first (7)
I believe the answer is:
dogcart
'carrier' is the definition.
(type of horse-drawn carriage)
'moggy across river tail first' is the wordplay.
'moggy' becomes 'cat' (I've seen this before).
'across' means one lot of letters goes inside another.
'river' becomes 'r'.
'tail' becomes 'dog' ('dog' can be a synonym of 'tail'**).
'first' means one lot of letters go next to another.
'cat' placed around 'r' is 'cart'.
'cart' after 'dog' is 'DOGCART'.
'taking' is the link.
(Other definitions for dogcart that I've seen before include "Two-wheeled vehicle" , "Light (actually horse-drawn) vehicle" .)