La familia estuvo andando
I am trying to say - the family was walking.
la familia estuvo anando en la playa - the family was walking on the beach.
How close am I ?
la familia estaba caminando en la playa - was walking (unspecified amount of time)
la familia estuvo caminando en la playa - walked (done and over with action)
use andando when saying strolling - leisurely
por la playa - would mean through the beach la familia is singular, never use ellos/ellas/ustedes form!!!
La familia anduvieron en la playa.
The family walked on the beach.
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La familia estuvo caminando por la playa.
Andando is also correct but is not commonly used in this case.
"Andando", but whether you use the imperfect or the preterite depends on the context. In other words, we need more context.