More than 600 Labour councillors have declared their full backing for all rail, postal, dock and other workers on picket lines, as the party comes under continuing pressure over its stance on strikes.

Amid a wave of strikes over pay and increasing talk of coordinated industrial action this autumn, the councillors signed an open letter saying they offered “full solidarity and support” to the Rail, Maritime and Transport workers’ union (RMT), the Communication Workers Union (CWU), Unite and others.

The latest workers to go on strike are CWU postal workers who rejected a 5.5% pay rise in return for changes to their conditions, while the RMT is planning a further round of rail strikes in the autumn. There are also 1,900 dockworker members of Unite at Felixstowe, the country’s biggest port, on an eight-day stoppage.

However, the issue of strikes is a vexed one for Labour. Keir Starmer, the party’s leader, has expressed sympathy with the aims of striking workers, but the party’s whips have asked frontbenchers not to appear on picket lines. The party is also stressing the need for negotiated solutions, with Starmer saying he “completely understands” why workers are striking but adding that his party would make sure talks were properly conducted.

Starmer’s equivocal stance on strikes has annoyed Labour’s trade union funders. Sharon Graham, the Unite general secretary, told the Observer earlier this month: “There’s no point giving money to a party that is basically sticking two fingers up to workers. It’s almost like an abusive relationship.”

Ahead of the Trades Union Congress in September, senior union officials have begun to talk more of scheduling strikes for the same time for maximum impact or staggering them for effect.

Earlier this week, Mick Lynch, the RMT general secretary, said: “We need a summer of solidarity, and a spring of solidarity if it needs to go through next year. The CWU, Unite, GMB, RMT and the others, we have to call on the entire movement ... to come into this action, to get members motivated and call them to the flag and vote yes for a wave of industrial action across the UK and internationally if that is what it takes, because we need to redress the balance in society.

“And not be dictated to by people from Eton and Harrow, telling us we have to give up our wages and give up our place. We are not going to have it.”

The joint letter was organised by the councillors Aneesa Akbar, from Hull, Jumbo Chan, from Brent, and Matt White, from Haringey, who are all active in the trade union movement.

In it, the councillors said they were “proud to witness this emerging renaissance of trade unionism, and we offer to the RMT, the CWU, Unite and many other trade unions taking action our full solidarity and support, on the picket line and elsewhere”.

They criticised the “obscene” situation in which millions of people in Britain were struggling to make ends meet and workers were told a “tsunami of lies” to keep workers in their place, while there were more billionaires than ever, and the profits and dividend payouts of Britain’s biggest companies soared.

The councillors said the actions of trade unions were a “welcome, important rejection of this dismal absurdity”.

“We urge all ordinary working people to take control of their destinies, and to join a trade union today,” they said.

Starmer was told to “get a spine” and send a “strong message” that Labour supported striking workers by Graham of Unite.

She told BBC Radio 4’s Broadcasting House programme “you cannot defend workers by being silent” and that neither Labour nor the Conservatives were aware of “how bad it is for people out there”.