Q:

Ending for gerunds

Elle va par les routes, rêvant.
Ces beaux oiseaux, je les ai vus volant.

In these examples of using a gerund (modified from here), is it necessary to change the gerund ending to match the noun, since the gerund functions like an adjective? In other words, should we use rêvante and volants?

A:

In French, gerunds are considered to be en VERB-ant while VERB-ant is called present participle. 
The two sentences you discuss here are marginal examples but these alternatives would be natural: 

Elle va par les routes, rêvant à ses amours.
Ces beaux oiseaux, je les ai vus, volant dans le ciel bleu.

Gerunds and present participles can be negated with ne contrary to the deverbal adjective:

Elle va par les routes, ne rêvant à rien.
Ces beaux oiseaux, je les ai vus, ne volant jamais ensemble.

Both gerunds and present participle are invariable verbal forms different from the deverbal adjectives rêvant/volant. Some French verbs display an orthographic contrast between the two:

CONVAINCRE convainquant (PRS_PART), convaincant (ADJ)
FATIGUER fatiguant (PRS_PART), fatigant (ADJ) 

